MINIMUM MEAN-SQUARED ERROR ADAPTIVE ANTENNA ARRAYS FOR
DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS SYSTEMS
By
JOHN EARLE MILLER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULRLLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1996
UNIVERSITY OF FLORIDA LIBRARES
Copyright 1996
by
John Earle Miller
This work is dedicated to my wife Kim, my children Marissa and Christopher and my
parents Frank and Garnet.
ACKNOWLEDGMENTS
I would like to thank all of the members of my committee for their involvement,
guidance and influence in this research. I would like to extend a special thanks to my
chairman. Dr. Scott L. Miller, for his support, many insights and thoughtful suggestions
which have influenced this work. The financial support provided by the Department of
Electrical and Computer Engineering is also gratefully acknowledged.
IV
TABLE OF CONTENTS
page
ACKNOWLEGMENTS iv
ABSTEIACT vii
CHAPTERS
1 INTRODUCTION 1
Direct-Sequence Spread-Spectrum with Multiple- Access 3
Adaptive Antenna Arrays 7
Previous Work 13
Dissertation Outline 21
2 A MULTIUSER ANTENNA ARRAY PROCESSOR
STEADY-STATE PERFORMANCE 24
The Multiuser MMSE Processor 25
Spatially Orthogonal Users 26
Nonorthogonal Users 30
Single-User System 34
One Weak User, One Strong User 34
Two Equal-Power Users 36
Two Strong Users 36
Numerical Results 37
3 THE MULTIUSER ARRAY PROCESSOR
ADAPTIVE PERFORMANCE 42
Mean-Based Performance Measure 43
Variance-Based Performance Measure 47
Simulation Results 49
4 BASE STATION RECEIVER PERFORMANCE USING A
SMART ANTENNA ARRAY IN A DS-CDMA SYSTEM
WITH IMPERFECT POWER CONTROL 52
System Description 53
Analysis 54
Simulations and Results 61
Conclusions 65
5 BASE STATION RECEIVER PERFORMANCE WITH A
SMART ANTENNA ARRAY IN THE PRESENCE OF
MULTIPATH FADING AND SHADOW FADING 68
A Single Cell with Signals Subjected to Rayleigh Fading 69
Multiple Cells with Signals Subjected to Rayleigh Fading
and Shadow Fading 7j
Analysis 76
Results gQ
Conclusions/Discussion gg
Summary g^
6 BASE STATION ANTENNA ARRAY
ADAPTIVE PERFORMANCE 9I
Signal and Channel Model 92
Adaptive Receivers 94
Simulations 9-7
Conclusions/Summary jq2
7 SUMMARY 106
Areas for Future Work 109
APPENDIX
A THE OUTPUT SNR PERFORMANCE SURFACE 1 13
LIST OF REFERENCES II7
BIOGRAPHICAL SKETCH I29
vi
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MINIMUM MEAN-SQUARED ERROR ADAPTIVE ANTENNA ARRAYS FOR
DIRECT-SEQUENCE CODE-DIVISION MULTIPLE- ACCESS SYSTEMS
By
John Earle Miller
August, 1996
Chairman: Dr. Scott L. Miller
Major Department: Electrical and Computer Engineering
This dissertation examines the performance of direct-sequence code-division multiple-
access receivers which use a minimum mean-squared error adaptive antenna array as a
predetection spatial filter. The array attenuates multi-access interference prior to
conventional, direct-sequence matched-filter detection. Conventional detectors are
vulnerable to heightened levels of multi-access interference. Minimum mean-squared error
processing seems like a natural choice for optimization in a code-division multiple-access
environment since several efficient search algorithms exist which are compatible with
decision-directed equalizer structures.
Two array structures are examined. The first suoicture uses a single set of array
weights to equalize more than one desired signal. For two strong incident signals of
unequal power levels, the steady-state response gives output signal-to-noise ratios that are
vu
leveled to a value near the maximum response of the weaker of the two users. The steady-
state adaptive performance of the multi-user array based on a least-mean-squares
algorithm is also examined. It is found that the maximum allowable step size based on
stabiUty arguments will also give adequate output signal-to-noise ratio performance of the
multi-user processor.
The second structure uses a single set of weights per multiple-access signal and the
array output feeds a conventional detector. The array/detector performance measure is
outage probability and outage-based capacity as a function of the number of array
elements and the degree of power control error. A robust, incremental measure of
performance~the per-element capacity-is defmed as the capacity per array element for a
given outage probability. Steady-state performance is evaluated for the case of directional
signals as well as for the case of signals subjected to multipath fading and shadowing. The
adaptive performance of the recursive least-squares algorithm is also investigated.
viii
CHAPTER 1
INTRODUCTION
Wireless communication based on direct-sequence spread spectrum (DS-SS) has
received considerable attention as an efficient signaUng format for code-division multiple-
access systems (CDMA). While DS-CDMA has a long history of use in defense
applications where jamming resistance and security are primary concerns,' some
proponents maintain that it will also serve as a high-capacity format for the next-
generation mobile-cellular and wireless systems for commercial use.^'^"*' System
specifications for domestic use have been endorsed by industrial agencies,* the results of
field trials have been pubUshed'* and it has been studied as a possible format for third-
generation mobile radio systems in Europe.'-'"
Despite a frenzy of activity and interest, DS-SS has a serious drawback: the near-far
effect This occurs when strong signals overwhelm a weaker desired signal during the
detection process. In a commercial mobile-ceUular system the near-far effect can occur at
the ceU-site base-station receiver. Incident signals originating from multiple-access, fixed-
power transmitters geographically distributed throughout a ceU can have incident power
levels that change drasticaUy as the transmitter positions vary. Signals originating from
transmitters near the base station may overwhehn received signals from transmitters on the
fringe of the cell. Vigorous proponents of commercial DS-CDMA systems have developed
transmit power-control formats to adjust each user's RF transmit power in real time so
that power levels of all multiple-access signals incident upon the base station are
approximately equal. Initially, it was stated that power control would need to be able to
adjust the transmit power over a 80 dB dynamic range in a mobile ceUular scenario." Field
tests indicate that 50 dB is more likely.^'^
Another use for DS-SS is in the Global Positioning System (GPS). Originally
implemented as a positioning system for the Department of Defense, it has enjoyed
commercial appUcations in mapping, navigation, and surveying." The world-wide GPS
uses a DS-SS signaling format to obtain relatively accurate estimates of geographical
position. An earth-bound GPS receiver determines its position by measuring the path
delays of several DS-SS signals which originate from earth-orbiting satellites. The near-far
effect is not a critical issue for many commercial GPS applications, such as aviation or
shipping, since Uie signals are subjected only to free-space path losses. Some commercial
appUcations, such as surveying, suffer other forms of signal losses, such as multipath
fading or shadow fading, and these can indirectiy lead to near-far limited performance.
This work examines receiver performance when an adaptive antenna array is used at
tiie DS-SS receiver. In such a system Uie array would act as a front-end spatial filter which
preserves the integrity of tiie desired signal or signals, attenuates interferere and
supplements the existing power conti-ol algoritiim. Such an approach might be compatible
with existing cellular CDMA standards. The remainder of tiiis chapter is divided into
several sections. The next section wUl give a quaUtative review of DS-SS systems. The
second section will review quantities and expressions used in the analysis of the minimum
mean-squared-error (MMSE) beamforming antenna arrays. The Uiird section wUl review
previous work germane to tiie research presented here and tiie last section will provide an
outline of the remainder of this dissertation.
Direct-Sequenc e Spread- Spectrum with Multiple-Access
Generating a DS-SS waveform involves multiplying a modulated baseband waveform
by a psuedorandom sequence sometimes known as a pseudonoise (PN) code. In an
asynchronous multi-access channel the incident DS-SS waveforms may be subjected to
random time delays and carrier phase delays and corrupted by noise. For a single-channel
receiver the multiple-access incident signals may be modeled by the expression:
Yit) = Relf^A„c„{t-r„)b„{t-T„)cxp{j{(o,t + d„)) + n{t)\ (1.1)
lm=l
where the index m refers to the mth of K signals. The quantity Am is the incident signal
level of the mth signal, Cm(t-tm) represents the mth PN code waveform which consists of a
sequence (of length Nc) of psuedorandom square pulses, (cm e {+1,-1 }) each with interval
Tc The quantity b4t-Tm) represents independent binary phase-shift keyed (BPSK)
modulation with equally-likely symbols (fc„ e {+1,-1 }) obtained via ideal, square pulses.
The time delay Tm is uniformly distributed over a symbol interval [0,Tb). The time delay t„
arises because the channel allows asynchronous access; users may begin transmission at
any time. The carrier frequency is designated by cot, and 0„ is the random carrier phase
uniformly distributed over [0,27t). The quantity n(t) is complex additive white Gaussian
noise (AWGN) with a one-sided power spectral density of No.
Note that in contrast to systems which use frequency-division multiple access
(FDMA), all users in a CDMA system share a common frequency cok. Frequency-division
channels are achieved by separating the carrier frequencies of multiple, bandlimited signals
to the point that they do not interfer with one another in the receiver. Code-division
channels in CDMA systems, on the other hand, are achieved through the low
crosscorrelation properties of the individual PN sequences. The PN sequences have many
interesting properties, including low crosscorrelation''* defined as
^» (^) = J c„ (f )c, {t + T)dt «Nc V m vt n. The limits of the integral are from zero to
NcTc. In a CDMA system the individual users share a common carrier frequency but are
assigned distinct PN sequences that permit code-division channeled Unks to a base station
or other receiver.
This woric assumes that detection of DS-SS signals is accomplished by a correlating
detector also referred to as a conventional detector. The ith user's DS signal is multiplied
by a synchronized replica of its PN sequence, passed to an integrate-and-dump filter and
Figure 1.1 Conventional detector, baseband model.
hard-limited to provide an estimate of the modulation symbol, as shown in Figure 1.1.
The conventional detector provides optimum performance for a single user in AWGN
but gives sub-optimum performance in a multi-access environment The performance of a
conventional detector in a multi-access environment depends roughly on the processing
gain which is a measure of the interference rejection capability of the detector and is
sometimes defined as the number of code symbols (or "chips") per modulation symbol (Nc
= Tb/Tc). For a fixed, finite processing gain it is always possible for strong interferers to
overwhelm the detection of a weaker signal and cause poor performance. The complex
baseband output of the integrate-and-dump portion of the conventional detector devoted
to user 1 consists of a desired signal component as well as multiaccess intefercnce and
noise components:
So.i{t) = AMt)+Z,K \c,{t)c„{t-T„)b„{t-T„)dt+ \ci{t)nit)dt
m=2 I
K
= AMt)+^A„[b„__,R„,{T„) + b„^,R„,{z„)\+ \c,it)n{t)dt
Kl{'^n,)=Jc„{t-tJc,(t)dt
(1.2)
where: °
T-
where bit 1 of user 1 is the detected signal of interest and T; = 0. The quantities R„ ^ (t„ )
and ^„ 1 (t„ ) are the partial crosscorrelation functions. The quantities bm,.j and bm.o
represent the contributions of the mth user's two bits which overlap the current bit interval
of the desired user. The values of crosscorrelations depend upon the sequences and their
relative time delays. The multi-access bit-error performance of the conventional detector
has been studied extensively; the dependence of the detector performance on PN-code
parameters has been well characterized.''-'* If one or more crosscorrelations are non-zero
and the corresponding input levels are large, those components may overwhelm the
desired signal in the detector. This is the near-far effect.
In mobile cellular or personal communication systems, signals incident on the base
station (also referred to as tiie uplink signal patii) can take wide-ranging incident power
levels because of the random placement of users about the base station. The current
solution to this dilemma is to vary the transmitter power levels in real time via transmit
power control. In a system using closed-loop power control the base station receiver
monitors the detected power level of each multi-access signal. As the incident power
levels of the multi-access signals vary because of changing channel conditions, instructions
are sent from the base station to the mobile units, via an uncoded narrowband side
channel, to individually increase or decrease transmit power. The aim of the power control
is to force the effective bit-energy-to-noise special density ratio i(Eb/No)S of all the
incident signals to the same threshold. In practice, however, the power control is unable to
perfectly track changing channel conditions because of doppler-induced multipath fading
and finite power control step size. Error in the power-controlled signal results in an
effective bit energy to noise specu^ density ratio {(Eb/NoUff) at the detector output which
is approximately lognormal in distribution.'^'* Field trials indicate that high-mobility
mobUe users (such as in fast-moving automobUes) have (Et/NoUff standard deviations of o,
= 2.5 dB while lower values (o; = 1.5 dB) have been recorded for low-mobUity users
(pedestrians).'
Existing cellular CDMA specifications also have a contingency for open-loop power
control.^ In an open-loop power-controlled system the mobile or personal units use the
detected base-station carrier power as a reference to adjust their transmit power. No side
channel is required. Some researchers have proposed estimation algorithms which would
allow a mobile to make estimates of the base station carrier power. The researchers
present evidence that the carrier-measurement error in terrestrial mobile cellular systems is
weU-approximated as lognormaUy distributed with a or,„ < 3 dB for vehicle speeds of up
to 60 mph and 8 dB of shadow fading. '^ Other researchers working in the field of mobUe
cellular systems based on low-earth-orbiting-satellltes (LEOS) have also found open-loop
power control error to be well-approximated by a lognormally distributed random variable
(r.v.).'"-''
If we generalize the previous work and assume that power control error may be
approximated by a lognormal r.v., then equation (1.1) may be rewritten slightly:
Yit) = Rei^ApO''-^c„{t-z,)b„{t-T„)exp(j{coj + e„)) + n{t)\ (1.3)
where e„ is normally distributed with zero mean and variance equal to Oj,^^ (N(0, o^J )).
Note that in previous works the lognormal approximation of power control error applied
to the incident power levels for open-loop power control, and to the (Et/No)^ for the case
of closed-loop power control. To simplify analysis, this research will assume that power
control error results in lognormally distributed incident signal levels for closed-loop or
open-loop power-controlled systems.
Adaptive Antenna Arrays
This research assumes an adaptive antenna array is used as a front-end spatial filter to
attenuate direction-dependent cochannel interference prior to conventional detection of
multi-access DS-CDMA signals. The anticipated role of the adaptive array wUl be to
attenuate the strongest interferers and thereby lessen performance degradations due to the
near-far effect. The antenna arrays explored here will be limited to beamforming arrays -
linear combiners - which use a minimum mean-squared error (MMSE) optimization
criterion. This technique is also referred to as Wiener filtering^^ or optimum combining."
A MMSE optimization is well suited to a mobile DS-SS-CDMA scenario since each user's
PN sequence, required for DS-SS detection and decoding, can provide a convenient
reference waveform. There are efficient MMSE adaptive algorithms which would require
no more side information than the code sequence and its timing.^^
The array is composed of ideal isotropic sensors with no mutual coupling between the
elements. Incident signals are spatially sampled by the sensors as they propagate across the
array. The incident signals are assumed to be narrowband. This approximation is valid for
spread spectrum signals as long as signal bandwidths are a small percentage of the carrier
frequency. Current specifications* call for bandwidths of 1.2288 MHz at a carrier
frequency of 850 MHz. The resulting double-sided signal bandwidth is much less than one
percent, so the narrowband approximation should be acceptable.
The physical displacements between the array sensors induce relative phase shifts on
our spatially sampled, narrowband signals. The phase-shifts are constant over frequency
and depend upon the array geometry and the signal's direction-of-arrival (DOA). The
complex signal component outputs of the A^ discrete antenna elements for the mth incident
signal can be described by a vector:
Sc^ = .?«(Oexp(y(/:y„ - (oj)) = ^„(Oexp(- jcoj)
where:
exp(y^„.o)
exp(y0„,i)
.exp(70„.^_,)_
(1.4)
u„=[exp(;0„,) exp(y0„,) ... exp(y0„^_, )]""
where the vector y„ (see Figure (1.2)) represents the physical displacements between
some reference point and the A^ array elements. An element of y^ (y„^n n = 0...N-1)
represents the physical distance between a reference point and the corresponding array
element The parameter k is the free-space wave number of the incident plane wave {k = 2
k/X ). Carrier phase-shifts have been absorbed into the complex, baseband function
gjt).lihe vector Un, contains electrical phase shifts resulting from the physical
displacements yn, between the elements and the point of reference. The individual elements
of Un, ( i.e. (j>m4, ) represent the mth signal's phase shift between the reference point and the
nth array element. The vector Um will be referred to as the interelement phase-shift vector
or simply the phase-shift vector and will not be considered a function of time unless stated
otherwise. For the remainder of the report constants will appear as normal block or script
characters, vectors will be represented by lower-case boldface and matrices by upper-case
boldface.
A diagram illustrating some of the physical quantities is shown in Figure 1.2. An
incident signal propagates as a plane wave s(t).The signals and noise are spatiaUy sampled
by each element of the array; ignoring an explicit time dependence allows the samples to
be represented by x„, n = 0...N-1, and form the input vector x. The inputs are weighted
(via the weight vector w = [wo w, ...wn.,]"^) and summed to form the beamfomer output.
The quantity ky is the phase-shift of the propagating wave due to physical displacement y.
As tiie wave propagates across tiie array tiie phase-shifts due to displacement give rise to
interelement phase-shifts which are contained witiiin Uie vector u. The lower part of tiie
figure illustrates how tiie direction-dependent phase-shift arises between two adjacent
elements. The quantity (j) is tiie phase shift between any two adjacent elements.
A change in array geometry would affect only tiie functional form of tiie relative
phase-shifts between tiie array elements. The quantity = 2nd(sine)/X in Figure 1.2 above
10
is the direction-dependent phase-shift between any two adjacent array elements for a linear
array. Arrays composed of individual elements arranged in a circle will be examined in
later chapters because, unlike linear arrays, they can resolve incident signals over the
direction-of-arrival interval [0, In). The phase-shift between any two adjacent elements is
given by ^ = (7uicos[6-2ji/N])/(Xsin[7c/NJ) for a circular array.
plane wave:
For a propagating plane wave s(t)
for a linear
array :
y = dsin0
^ = ity = 2;r y / A
d = element spacing
the phase - shift (p
between the two
elements is then:
^ =
2;r dsin0
Figure 1.2 A directional plane wave incident on a linear array of sensors.
The baseband array output vector is Uie sum of Uie noise and signal vectors:
^ = nW + Zs™{t) =n{t) + f^gjt)u^ (1.5)
m=l
m-1
11
where n(t) is the AWGN vector. The noise is spatially and temporally white. The
narrowband portion of the array autocorrelation matrix may be expressed as a sum of
outer products or as a product of matrices:
^ = E[x'x^]=Gll + f^Alulul =all + V'A'V' (1.6)
m-l
where U is the matrix of phase-shift vectors and A is a diagonal matrix of input
amplitudes, (•)' represents a complex conjugate and {•f a vector transpose, respectively.
The expectation £[•] averages the AWGN, n(t) which is assumed to be N(0.a,^). The
norm-squared of Um is:
ikir=">™='^ (1.7)
independent of signal DOA where (•)" represents the hermitian transpose.
The MMSE weight vector minimizes the mean-squared error between the processor
output and the reference signal. The weight vector is given by:
w„ = R"*p (1.8)
where p is the steering vector and is given by:
P = E[r{t)s'j] = AjAgUj (1.9)
and Sd and Ud are the signal vector and phase-shift vector of the desired signal respectively.
The quantity r(t) is the time varying reference waveform with a peak amplitude of A/?. The
MMSE weight vector maximizes the output signal-to-interference-and-noise-ratio"
(SINK) which is defined as the quotient of the desired signal power and the sum of the
interference and noise power SINE = Ps/(Pi+Pn). The output signal-to-noise ratio (SNR)
for the mth signal is the quotient of the output power of the mth signal and the output
12
noise power. It is given by:
27
ijn rorn- I m
|2
SNRo„ = -^ = 5A^/?/„ ' ; ,;' < N ■ SNRi„ (1. 10)
The quantity SNRi„ = A^Va^ is the incident SNR for the mth signal, Ps.m is the desired
signal output power, P, is the interference output power and Pn is the output noise power.
For a single signal in AWGN, Wo = u*, and the upper bound becomes an equality. The
choice of Wo = u^* represents the spatial equivalent of a matched filter (to the rath signal)
and is referred to as a conventional beamformer^'*, a classical beamformeT^\ or a
maximal ratio combiner}^
The sum total of the number of beams and nulls an array is capable of directing
simultaneously is N-1, one less than the number of array elements. This is also referred to
as the degrees-of-freedon?^ of an array.
At this point, it may be important to explain the difference between the terms
adaptive array and diversity combiner. The term adaptive array implies that the processor
uses an array of sensors and exploits the phase and amplitude shifts between the array
elements, induced by directional signals, to make processing decisions. A diversity
combiner, on the other hand, exploits some degree of statistical independence between the
input samples to ensure signal integrity at the summed output For example, in the case of
communication channels which contain multipath fading, an array of sensors might provide
statistically independent spatial samples. If a signal is faded at one sensor, it might not be
faded at anoUier sensor; the statistical independence of die inputs is exploited by die
processor to improve tiie integrity of the overall output response. There are many diversity
combining strategies.'' A popular strategy for analytical purposes, mentioned previously
13
as maximal-ratio combining, maximizes the output SNR but does not allow adaptive
interference suppression. The diversity-combining counterpart to MMSE filtering is
referred to as optimum combining. It is sometimes possible to achieve statistically
independent samples through time sampUng. Aldiough not an issue in this research, a few
examples of time diversity will be cited in the following section.
Previous Work
This section is a survey of previous work in adaptive beamforming antenna arrays and
in CDMA that is pertinent to this research. The first part of the survey will give a
historical perspective to research in adaptive antenna arrays. Details of the earlier work
oriented towards radar and military communications is admittedly sparse. As the timeline
and focus become more current the coverage will become more detailed. The second part
of the survey will review the most recent work in which adaptive arrays function as an
integral component of DS-CDMA systems. The last part of the survey wUl examine
previous work in cellular CDMA ( without antenna arrays) as it appUes to this research.
A beam-steered array was investigated by Applebaum^' but the results were not
published in open literature until a decade later.^° Widrow et al.^' reported on an antenna
array using a least-mean-squares (LMS) processor in 1967. A special issue of the IEEE
Transactions on Antennas and Propagation devoted to active and adaptive antennas was
pubUshed in 1964,'' 1976'' and 1986.^^ Other processors were investigated by Frost'',
Griffiths'^ and Schorr." Much of the initial adaptive array work focused on the
performance of particular processors or radar-oriented appUcations. One exception to this
was the maximal ratio diversity combiner investigated by Brennan.'*
14
Subsequent work explored the role of adaptive arrays in communication systems.^*' '^^
In particular, Compton'^ presented a qualitative evaluation of an experimental adaptive
array in a DS communication system. The evaluation focused on a single desired signal
and a limited number of jammers. One of the conclusions reached by Compton is that the
array makes appreciable contributions to interference suppression. Winters'" studied the
acquisition performance of an LMS adaptive array in a DS system using four-phase
modulation with two PN codes, a short code for rapid acquisition and a long code for
protection against jammers. Ganz'*^ evaluated the bit-error-rate (BER) performance of a
receiver employing an adaptive array and one of several detectors for binary-phase-shift-
keying (BPSK), quadrature phase-shift keying (QPSK), or differential phase-shift keying
(DPSK) modulation. The receiver was subjected to continuous-wave (CW) jamming.
Several authors have investigated the use of antenna arrays for mobile or personal
communications. Bogachev and Kiselev'*' evaluated optimum-combining diversity arrays
for the case of a single interferer. Winters" conducted a comparative study of optimum
combining and maximal ratio combining base station arrays in a multiuser mobile
environment with multipath fading but no shadow fading. The results quantify the possible
improvement in SINR if optimum combining is selected over maximal ratio combining.
Like optimum combining, maximal ratio combining preserves die integrity of the desired
signal, but unlike optimum combining maximal ratio combining has no abihty to adaptively
suppress interference. Winters did propose the use of psuedonoise codes to generate the
LMS reference, but transmit power control was menuoned only for its effect on the
convergence properties of the LMS array. He did not investigate in any detail the
condition when the number of users greatly exceeds the number of array elements. Winters
15
also explored the use of adaptive arrays on base stations for in-building systems using
dynamic channel assignment'" He again considered a PN-coded PSK modulation and
circumvented power control considerations by assuming interferers were of equal power
and much stronger than the desired signal. A more recent study"' investigated the
acquisition and tracking performance of LMS and sample-matrix-inverse (SMI)
beamformers in a time-division multiple-access system.
Yeh and Reudink"*^ examined the contributions made by spatial diversity combiners to
specu-al efficiency in FDMA cellular systems when the arrays are located on the base
station and on the mobiles. Glance and Greenstein"^ also examined the contributions of
diversity order (die number of array elements) on average BER in a mobile FDMA system.
Vaughan'^ discussed the benefit of MMSE combining at the mobUe in an FDMA system
and concluded with the comment that for MMSE combining to be successful wide
bandwidth signals, such as those found in spread-spectrum systems, are necessary.
More recently, adaptive array research has examined commercially-oriented
applications as CDMA and non-CDMA wireless systems gain popularity. At this time,
antenna arrays are under investigation as a means of providing space-division channels in
multiple-access systems. The technique, called space-division multiple access (SDMA) by
some authors, utilizes the spatial filtering properties of tiie array to selectively receive
signals which share the same time slot and tiie same frequency band. The SDMA
technique might apply to eitiier time-division multiple-access systems (TDMA) or CDMA
systems. Swales et al."' established tiiat a steerable, multi-beam antenna array can increase
tiie capacity and spectral efficiency of a cellular system. Suard and Kailath'" studied tiie
upper bound of Uie information-based capacity of a wireless system which used a base-
16
Station antenna array in the uplink path. Experimental studies have been conducted. Xu et
al. and Lin et al.'^ have examined algorithms based on direction-finding techniques
MUSIC^^ and ESPRIT.^ They have concluded that SOMA techniques based on DOA
estimation will not be effective in multipath-rich environments. Xu and Li" developed an
SDMAADMA protocol. Ward and Compton**" examined the contributions that a
receiving array can make to the performance of an ALOHA system.
Arrays which include spatial and temporal adaptive processing nested within LMS
feedback loops were proposed by Kohno et al.'* and Ko et al.'' Kohno used an LMS array
in conjunction with adaptive temporal filtering which successively canceled DS signals due
to multi-access interference. Ko described a null-steering beamformer nested within an
LMS loop, not necessarily restricted to CDMA applications. Both considered limited
multi-access scenarios with deterministic interference parameters.
Diversity combining is not necessarily restricted to space diversity. Balaban and
Salz • examined in detail the performance of a general multi-channel MMSE combiner
working in conjunction with a decision-feedback equalizer. They estabUshed an upper
bound on BER which is a function of the MMSE. A great deal of work has also been
devoted to temporal diversity combiners which consist of a single input to a bank of
matched filters, the outputs of which are coherently combined via maximal ratio
combining. This is the basis for the RAKE" receiver as well as variations studied by otiier
autiiors. Several structures were examined by Turin^^ while Uhnert and Pursley"
examined diversity combining in multiuser CDMA system in which successive bits are
spread witii different PN code subsequences. Wang et al. consider diversity for an indoor
DS-CDMA system with Rician fading.*'
17
Some authors have studied the possible contributions made by arrays to the upUnk
path performance in cellular systems. Liberti and Rappaport have studied the effects of a
directive, steered-beam base-station receiving array on uplink performance in a CDMA
cellular system with perfect power control. The study focused on the effect of beam shape
and beam width on the average BER. It was found that beam width has the greatest
impact on performance and that adding a three-element array to the base station can
improve BER performance by three orders of magnitude. Tsoulos, Beach and Swales have
recently examined the role of adaptive antenna technology in large "umbrella" cells
overiaying smaller microcells in cellular CDMA systems^ ; they also examined the outage-
based capacity enhancement due to an adaptive antenna array using a recursive least-
squares (RLS) processor in a multi-ceU CDMA environment^^ The latter study was
confined to simulations of the total interference to calculate outage. The authors
concluded that an antenna with 6 dB of directivity gain can increase capacity by a factor of
five.
Winters, Salz and Gitlin^ studied the effects of optimum combining spatial diversity
arrays on the capacity of a TDMA system in which the total number of incident signals
was less than or equal to the array DOE. They applied previous work^-^' which resulted in
an upper bound on BER, The assumption of high input SNR allowed a zero-forcing
approximation to the optimum combining solution. Using these assumptions, and
examining analytical expressions for BER, they found that optimum combining with N
antennas and K interferers gives the same results as a maximal ratio combiner with N-K+1
elements and no interferers. Their theoretical results, as they point out, no longer apply
when the number of interferers exceeds the number of antenna elements.
18
A structure which combines the spatial filtering of an antenna array and the temporal,
diversity-combining properties of a RAKE receiver have been proposed and investigated
by Khalaj in concert with several other authors.^"'"'^' The structure is intended for use in
channels with frequency-selective multipath fading. The structure allows the resolution of
identifiable multipath rays by the time-filtering properties of the RAKE combiner and by
the spatial filtering properties of the antenna array.
A number of authors have conducted brief simulation-based studies of adaptive arrays
for DS-CDMA systems. Yoshino et al." examine the simulation performance of two
RLS-based spatial diversity combiners operating in concert with (Viterbi) sequence
estimators. One processor subtracts estimates of the cochannel interference from the
output prior to estimation of the desired user's data sequence while the other processor
does not. Wang and Cruz" examine the pattern behavior and BER of a six-element arrays
based on the RLS and ESPRIT algorithms with six active users with well-separated
DOAs. Liu^" examined the performance of an LMS array with a scenario-dependent
matrix preprocessor which aids in interference cancellation. Hanna et al." investigated the
BER performance of a two-element LMS array which operates in conjunction with an
adaptive equalizer.
Perhaps the most in-depth study of the possible contributions of MMSE adapuve
beamforming arrays to the performance of mobUe ceUular CDMA systems using closed-
loop power control has been made by Naguib in concert with other authors. Initial work^*
focused on steady-state performance of an array of sensors, each foUowed by a DS
conventional detector, which functions as a post-detection combiner (termed code-
filtering by the authors). Analysis resulted in a simple expression for capacity. The signal
19
model was for unfaded signals originating in an isolated single cell. Power control error
was modeled by assigning a single interferer an incident signal level 10 dB higher than the
other signals.
Naguib, Paulraj and Kailath" extended the steady-state model in order to determine
the outage probability in a cellular system with BPSK modulation, perfect power control,
shadow fading, multipath fading, and cochannel interference equivalent to two tiers of
surrounding cells. Assuming that the array pattern response consisted of a main lobe and
no sidelobes resulted in a simple, closed-form expression for an upper bound on outage
probability which simplified to the single-channel results of Gilhousen et al." when the
array is reduced to a single element. Modeling the MMSE processor as a maximal ratio
processor and assuming the interference was Gaussian resulted in a simpler expression for
the outage probability upper bound.^* Naguib and Pauh^j then modified the simulation
model to include DPSK-modulated signals and determined the Eriang capacity." The
uplink performance widi M-ary orthogonal modulation was examined as well. *°
A recursive beamforming algorithm was proposed by Naguib and Paukaj*' and
simulated results were presented. The algorithm performed recursive updates on the
matrix square root of the inverse of the covariance matrix. The authors claimed that the
accumulation of numerical and quantization errors may cause the covariance matrix
inverse to cease being hermitian defmite and that updating the matrix square root will
allow the covariance matrix inverse to remain hermitian definite even when the matrix
square root is not. Thus, say the authors, numerical instabUities are avoided.
Naguib and Paulraj*^ continued their invesUgation of cellular base station antenna
arrays by examining the tradeoffs in coverage area, mobile transmit power, and capacity
20
that are available when an array is used and the users are subjected to perfect power
control. Using simplifying assumptions the authors derived expressions which generaUzed
the effect of the antenna array on performance. This research will attempt to extend some
of the results to the case of imperfect power control.
Later investigations have resulted in detailed, simulation-based studies of an IS-95
system which uses orthogonal signaling, forward error-correction coding, closed-loop
power control and a base-station antenna array. Unlike their previous studies Naguib and
Pauhaj appUed a model for imperfect closed-loop power control. They studied the
standard deviation of power control error, although where the error is defined is not clear
in the paper. Using simulation results they show that power control error dependence on
die power control step size, the number of array elements, and the maximum doppler
frequency and the spread in DOA of the multipath rays.*^ The dependence of BER on the
same parameters was the topic of a subsequent paper.^
A multiuser LMS array for use in GPS receivers was examined by Beach et al.*' in a
simulation study. The results were confined to plots which show die evolution of the
adaptive array pattern over time in the presence of CW jammers; no steady-state results
were presented. The near/far effect was not an issue in the study.
A variety of authors have investigated the performance of cellular CDMA systems
with imperfect power control and no antenna array. Simpsora and Holtzman** used
simplified analytical models in order to provide insight into the interactions between power
control, coding and interleaving. Stuber and Kchao'^ examined a multiple-cell CDMA
system and evaluated the dependence of BER as a function of Uie distance from die base
station. Jalali and Mermelstein** conducted a simulation study of a microcellular CDMA
21
system which included imperfect closed-loop power control and antenna diversity with
square-law combining. Milstein et al/' studied the average BER performance of a
multiple-cell system in which users where subjected to power control error which was
described by a uniform random variable. Newson and Heath'" examined a CDMA system
which suffered from imperfect sectoring and lognormally-distributed power control error
and made capacity comparisons to TDMA and FDMA systems.
Dissertation Outline
The remainder of the dissertation will be divided into six chapters. Chapter 2 will
examine some aspects of the steady-state performance of a multiuser MMSE array; some
limited aspects of an adaptive LMS version were investigated by Beach et al.*^ The
near/far performance will first be investigated for strong, spatially orthogonal users and
will then be extended to the case of two users with any DOA spacing. The analysis will
focus on the ability of the array to confine the signal outputs to the same output SNR. The
results will show that the array may be suitable when signals are well-separated in DOA
and do not outoumber the array DOFs. The analysis will borrow some of the analytical
techniques used by Gupta'' and apply them to evaluate the steady-state performance of
die multi-user MMSE array. Unlike either previous study, however, the analysis will focus
on the performance of the array in a near/far environment.
The third chapter will examine the adaptive performance of the multiuser MMSE
array when it is implemented as an LMS processor. Analysis will show that the rule-of-
thumb for picking step size to ensure stability also serves as a limit to reduce the spread in
output signal levels due to LMS misadjustment
22
The fourth chapter will examine the steady-state performance of an MMSE array
feeding a conventional detector. Unlike the array of the first two chapters, this array will
be dedicated to a single user among many multiple-access users. The single-cell users will
not be subjected to many of the influences normally found in a mobile environment, such
as multipath fading, shadow fading or interference which originates from surrounding
cells. This will simpUfy the analysis: the intent is not to provide an analysis fraught with
mathematical rigor, but to use simplifying assumptions to aid in the derivation of closed-
form analytical expressions which accurately predict the uplink performance and which
clearly show the dependence of uplink capacity on the system parameters, especially the
array and the degree of power control error. The performance measures will be outage
probability and outage-based capacity.
The analysis of chapter four will result in a simple closed-form expression for the
capacity which is linear in the product of processing gain and the number of array elements
and which decays exponentially with increasing power control error. The slope of the
capacity Une (the per-element capacity) will serve as a robust measure of the array's
incremental contribution to capacity. Although some of the assumptions are highly
idealized (i.e. no multipath fading), diey will allow some weU-understood quantities, such
as array gain, to be exploited in the analysis. Some of the work in this chapter is related to
that of Padovani^ and also to Naguib and Paub-aj.*^ The details wiU be discussed during the
chapter's derivations and discussions.
Chapter 5 will extend the steady-state multi-access model of Chapter 4 to include the
effects of multipath fading, shadow fading and outer-cell interference. The performance
measures will again be the steady-state outage probability and outage-based capacity.
23
The sixth chapter will examine the recursive-least-squares (RLS) adaptive
performance of the MMSE adaptive array in a mobile cellular scenario in a nonstationary
channel. The discrete-event simulation model includes time-varying multipath fading,
stationary shadow fading and time-varying outer-cell interference.
The last chapter will give a brief summary and some conclusions of this research. It
will also identify some areas of future work.
This research makes contributions in two areas. First, analysis and numerical solutions
provide more insight into the steady-state and the adaptive behavior of the multi-user
MMSE array (chapters 2,3) than was provided by previous authors. Analysis shows that
the array can remove die near-far effect and level the output SNRs to nearly the same
level, under certain circumstances. Second, the analysis and simulation of a single-user
array operating in conjunction with a conventional detector results in simple expressions
for capacity when the signal sources are subjected to imperfect power control (chapters
4,5,6), The results characterize the incremental contributions an adaptive array can make
to uplink performance. This is in contrast to previous works which examine the effects of
closed-loop power control error only in simulation for Umited scenarios.
i«'
CHAPTER 2
A MULTIUSER ANTENNA ARRAY PROCESSOR
STEADY-STATE PERFORMANCE
This chapter examines the ideal, steady-state performance of the multiuser MMSE
array proposed by Beach et al.*^ The authors conducted a simulation study of the pattern
behavior of a multiuser LMS adaptive array with and without directional interference.
They did not provide any analytical results which give general insight into the array's
steady-state performance. In related work, Gupta" examined a multi-user steered-beam
array for non-CDMA applications and simplified his analysis by assuming the special case
of spatially orthogonal users. Gupta was interested in using the array's effective aperature
to improve the output SNR of the desired signals and in using the adaptive properties to
reject interference. Other authors have shown that steered-beam arrays and LMS arrays
give die same output SNR performance as long as the steering vectors differ by a
constant.'^ Using diis rationale it might seem that Gupta's work could provide some
insight to this problem. However, he was trying to configure the array processor to avoid
the power leveUng effect that this work is attempting to exploit.
This chapter will focus on the array output SNR performance for strong incident
signals. The analysis will deemphasize the role of the spread-spectrum signalling other
than to note that it provides a relatively easy way to provide separation and detection of
the multiple, desired signals present at the single array output. The remainder of this
chapter will be divided into several sections. The first section will provide some qualitative
information about the multiuser MMSE array processor and give some refinements to the
24
25
general array equations given in the introduction. The second section will develop
analytical expressions of the output SNR for K <N-1 spatially orthogonal users. The third
section will present analytical expressions for two users which have arbitrary DO As and
are therefore not necessarily spatially orthogonal. The last section will compare the
analytical and the numerical results and discuss their implications.
The Multiuser MMSE Processor
The multiuser array treats each of the K signals as a component of a single composite
desired signal. This is accomplished via a reference signal which is a sum of the modulated
PN sequences of the desired incident signals. A block diagram is shown below in Figure
2.1. In the figure all functions of time are ignored. For analytical simplicity we will assume
that perfect estimates of the modulation and code waveforms are available for the
generation of the reference waveform. This will not usually be the case. Generation of the
reference signal can be a challenging issue and it has been investigated by other
authors."-^-"'
The array will force the outputs of the individual signals to the levels of that signal's
component of the reference waveform. To avoid the near/far effect the individual
components comprising the reference waveform have the same peak ampUtudes. Since all
of the incident signals share the same beamformer weights, the output SNRs will be
leveled to approximately the same value. The analysis will show that the array will force
the output SNRs to values very close to the maximum output SNR of the weakest user
imder certain conditions.
26
The steering vector defined by equation (1.9) represents the crosscorrelation between
the reference signal and the multiple, independent input signals. It becomes a sum of
single-user steering vectors:
P = SP™=/?XA„"m
(2.1)
where A„ and Um are the incident amplitude and phase-shift vector of the mth signal,
respectively.
V
Xo(t)
o
o
Output
XN-l(t)
Wo
o
o
o
<>
Reference
signal
e(t)
Wn-I
To a bank of
conventional
DS detectOTs
noit) + ^So„{t)
B=l
^.0 - T,>,0- T.)
I
^)^b,(t- T,>,0- T,)
Figure 2.1 Multiuser MMSE processor
Spatially Orthogonal Users
Consider the limiting case of multiple users which are spatially orthogonal to one
another (uj^i^ =0 V i^tk ). The minimum source separation (in DOA) which aUows
spatial ortiiogonality corresponds to tiie Rayleigh limit.'' The Rayleigh limit is generally
27
considered to be the minimum amount of source separation which still allows resolution of
two sources by a beamformer.^'* If the users are orthogonal and the array elements are
isotropic the inverse of the autocorrelation matrix is:''
R-' =
. ^ NSNRi„ . T
1-2. - "- »-
/IN. m-
Til+NSNRi,
(2.2)
and the resulting MMSE weight vector, Wo = R p, is given by:
w.
=-i-
^l ;^i+
A„
NSNRL
•u.
(2.3)
Substituting these quantities into the general expression for the n'th users output SNR
(equation (1.10)) gives:
SNRo. =
NSNRi„
l + NSNRi
\2
f K
■nj
NSNRL
^-^
^„til + N-SNRiJ%
(2.4)
Some coarse judgements on performance for multiple users can be made using
equation (2.4). If {SNRim » 1 V m =1 ... K} the output SNRs for all of the users are
approximately equal to:
r K
SNRo'^ N-
* 1
y^iSNRi„
(2.5)
If there are K users with the same input SNR = SNRi then equation (2.5) simplifies to:
NSNRi
SNRo =
K
(2.6)
which shows that die users will all have output SNRs equal to the maximum possible SNR
divided by die number of active users. If die array parameters remain constant, die
performance will degrade as the number of users increases.
If there are S users with the same input SNRi = kjSNR and K-S users with SNRi =
SNR then the output SNR for the K users smiplifies to:
SNRo = N ■ SNR ^ (2 1)
If kj-(K- S) » S (a possible near/far scenario) the expression for ou^ut SNR
simplifies further:
i N-SNR
SNRo
K ^ S
LNSNR (2.8)
' K = S
K
where:
S = no. of strong users with input SNRi = kj SNR, ki » S
K = total number of users
J^ - 5 = no. of weaker users with input SNRi = SNR
From the equation immediately above it is apparent that one weak user {K-S = 1) will
dominate the overall performance of the array. This might make intuitive sense, even for
the general case. A minimum MSE processor wiU try to force each output signal to have
the same value as that signal's component of the reference waveform. This strategy will
favor weaker incident signals which need large weight values to cause their array output to
match their component of the reference waveform ( i.e. minimum error power). Large
incident signals will be attenuated via phase shifts and the noise power - the denominator
of equation (2.4) - will be large thus causing a lower than maximum output SNR even for
strong input signals.
Equation (2.8) above establishes that the weakest incident signal drasticaUy affects die
overall performance of the array. The worst array response towards an arbitrary signal
(SNRon) for a different input signal (SNRifn) can be found by finding the value of SNRi^
29
which forces dSNROfJdSNRif^, the derivative of the nth signal's output SNR with respect
to the mth signal's input SNR, to zero:
dSNRo^
dSNRL
NSNRi„
\\ + N-SNRi„
r, XT rxro. \ /" D
l-NSNRL
l + N-SNRi
m J
NSNRi,
\-2
fiil + NSNRi,)^
=
(2.9)
From the equation above its is evident that a minimum occurs for SNRi^^i = 1/N.
Substituting SNRifn = 1/N into equation (2.4) gives (for SNRi^ » l,n^m):
SNRo, =4-
SNRo^ = 1
^ NSNRi„ Y
^l + N-SNRi^^
-4
(2.10)
For SNRifn = 1/N the value of tiie mth eigenvalue of R is twice tiiat of tiie noise-only
eigenvalues. At this input level the array is barely able to resolve the m'th input signal
from the input noise.
It might be useful to define an output SNR "spread" which bounds die output SNRs
for aU of tiie users. From equation (2.4) it can be seen tiiat tiie largest and smallest input
signals result in the largest and smallest output SNRs respectively. We can predict tiie
output spread by tiie difference between tiie largest and smallest output SNRs:
^SNRo = SNRo^^ - SNRo_
^ NSNRL., V f NSNRU V
{l + N-SNRi„
{l + N-SNRi„
^ NSNRi„ Y
ha + N-SNRij'j
(2.11)
If we let SNRimax pass to infinity the output spread becomes:
lim
30
1 ( NSNRi^^ Y
[l + N-SNRU
^ NSNRij
(2.12)
The equation above simplifies to:
(2.13)
ASNRo = ^_±y^lSNRi^
An upper bound on the output SNR spread results:
ASNRo < 2 + i (2 U)
NSNRi^^
miD
In summary it would appear that for N-SNRim » 1, m=l...D, the user outputs will
be near the value of the maximum output SNR for the weakest user and the output SNR
spread will be equivalent to 2.
Nonorthogonal TTsp.rs
Analyzing the general case of multiple narrowband users which are not spatiaUy
ortiiogonal is difficult because of the matrix inverse in equation (1.8). When tiie users are
not orthogonal the analytical matrix inverse (via the matrix inversion lemma) quickly
becomes intractable as the number of users increases beyond unity. The general case of
only two users was examined in detail. The approach taken here is to derive expressions
for the two output SNRs. The expressions are tiien interpreted as transfer functions with
poles and zeros which are dependent upon Uie input levels and tiie DOAs. Critical points
of the functions are then evaluated.
31
The autocorrelation matrix R for two independent, narrowband users in AWGN
(equation (1.6)), is given by:
= o^[l+ SNRiyy, + SNRiyy^]
The inverse of R (R-1) may be calculated via die matiix inversion lemma:
D-i _ 1 T 1 SNRii . T
(2.15)
(j; \ + SNRi,
'i"i
SNRL
' 1 + NSNRL "
(2.16)
u]u\SNRL
l + NSNRL '
I + SNRL-
T •
SNRL SNRL
1 + SNRi^
The steering vector p which results from equation (2.1) is:
P = /2A,u; + ^u; = (J„R^SNRiy, + a^R^fsMI^u, (2. 17)
Substituting equations (2.16) and (2.17) into equation (1.8), wo = R'^p, gives tiie MMSE
weight vector:
"'"^^'^^'^o [^/^^(l + S^^h ) - uJulSNRi, 4SNRi;]a[ +
R fi , , (2.18)
— ySNRi, (1 + SNRi, ) - u]u[SNRi, V^AWTju;
The quantity D^ is defined by:
A =l + N(SNRi, +SNRi,) + [N-\uJul\yNRi,SNRi, (2.19)
Two vector inner products are required to find the output SNR defined in equation
(1. 10). If we assume tiie point of reference for the array is die same as the physical center
of the array, Uie imier products will be real.^^ An inner product between two phase-shift
32
vectors may then be expressed in dot-product form: uj"u* =Ncosa^^ where a^ is the
angle between the two vectors in signal space. Substituting this relationship into equation
(2.18):
II ||2 R^ , . .
Ko i = -^ ^[SNRi, + SNRi^ ) + 2N^SNRi,SNRi^ cosa,^ +
n o
2
R
^^N'SNRi.SNRi, sin' a,,{4 + NSNRi, + NSNRi, - 2N^SNRi,SNRi, cosa.J
o'„^.
where: />„=! + NSNRi, + NSNRi, + N' SNRi,NSNRi, sin' a,^ (2.20)
Expressions for the numerator of equation (1.10), the output SNR, are possible for the
two users:
I |2 /?' r
r>o\ =-ZTJ^[^yfsmI{l + NSNRi,,sm' a,,)+N,fsm;;cosaJ (2.21)
Substituting equations (2.20) and (2.21) into equation (1.10) wiU result in the steady-state
output SNR for the two users at the point of minimum mean-squared error.
An additional substitution might simplify the expressions further. If we substitute the
expression SNRij= a^ SNRi2 into equations (2.20) and (2.21) the output SNR can be
expressed as a quotient of polynomials in a with coefficients that are functions of NSNRi2
and ai2. The numerators of the quotients are:
num, = N'SNRi^ll + NSNRi^ sin' a,^f
cosa,-
n2
l + NSNRi^sm^a^2j
num. = N'^SNRi^, sin^ a
2 , cosa,2
(2.22)
n2
NSNRi^ sin ' a^^ NSNRi^ sin ' a .^
for user 1 and user 2 respectively. The denominator of both biquadratic temis
33
Denss], = a^N^SNRi^ sin' a,,[4 + NSNRL,{a^ - laco&a,, + 1)] +
NSNRi^la'' + 2acosa,j + 1)
and the expressions for the two output SNRs become:
(2.23)
SNRo, = a'SNRi, ^^^^ SNRo, = SNRL -^^^^^^ (2.24)
^^^sm Denssu
The expressions for output SNRs may be interpreted as transfer functions in the
variable a. The numerators may be treated as products of second order polynomials with
easily determined roots which may be interpreted as real zeros of the output SNR "transfer
function." The roots of the numerators are:
cos a,
z =0 ^12
1 + NSNRL sin' a
12
(2.25)
_ cosa,
"Bum 2
>sci:,, r I -. 1
oA/CA/icT^ V±^^^-^NSNRi,xm'aJ
INSNRi^sm a^^y 2 i2j
The denominator is a fourth-order polynomial in a and analytical expressions for its roots
are not particularly useful in the general case. To circumvent this difficulty the analytical
expressions for output SNR will be examined and approximated for specific cases of a^
(the near-far ratio) and 0^2 (the angle between the phase-shift vectors in weight-space)
and results will be compared to numerical solutions. Note that ai2 is determined
exclusively by the users' DOAs for a given antenna array configuration. The output SNR
expressions will be examined for four scenarios of the variable a:
^•^ =0 (single user system)
2. < a2 « 1 (1 vitdk, 1 strong user)
■^- ^ = 1 (2 strong, equal power users)
4. 1 < a2 < 00 (2 strong, unequal powers)
34
Two cases of DOA separation will be examined for each of the four scenarios listed
immediately above. One case (referred to as case a) will be that of spatially orthogonal
users (cos2ai2 = 0, sin2ai2 = 1). This will allow comparisons witii the tiie results in the
previous section. The second case {case b) will be for DOAs with spacings greater than
the Rayleigh limit (0 < cos2ai2 « sin2ai2 < 1) and will also be referred to as well-
separated.
The following analysis will consider approximations to equations (2.22)-(2.25) for
some combinations of the intervals of a listed above. Ratiier tiian being exhaustive tiie
analysis will focus on critical points which will reveal trends in performance.
Single-User System
When a = we have a single user system and :
SNRo, = a^SNRi
Den
=
SNR
(2.26)
SNRo, = SNRL -^^^^^^ =
Den
NSNRL
SNR
One Weak User. One Strong U.ser
For the case of one weak user and a moderately sti-ong user which are well separated
(case b) die numerators and denominator of equation (2.24) can be approximated by:
num, = N^SNR^l + NSNRi^ sin' a,J
a + -
cos a,
12
l + NSNRi^sin^a^^
(2.27)
num^ '^N'^SNRi^ sin'' a^^
li
a' +
NSNRi^ sin' a,2
35
Derisffn = N^SNRil(4+ NSNRi2)sm^ a,^
a' +
NSNRi^ (4 + NSNRi^) sin^ a,^
If we can further assume that NSNRi2 » 4 then the output SNRs may be approximated
by:
SNRo, =a^N- SNRL sin^ a
[a + (cosa„)z^,f
[^'+^2^]'
SNRo^ = N • SNRi^ sin' a,2 7 t-
(2.28)
where: ^.^ =
P2b =
_ ^2*
NSNRi,sm'a,, ' *''" N' ■ SNRi', sin' a,, NSNRi,
Where the notation Zjb represents the numerator zero for scenario 2 and DOA case b.
Note that pzb < Zjb- Also note that for scenario 2, SNRoj is smaU since it is multipUed by a
second order zero at a = 0. As a increases, SNRoj increases with some possible wrinkles
in the magnitude curve added by the numerator zeros ( a = Z2bCosai2) and denominator
poles ( a = yfp^). Note that since the cosai2 term in the numerator of equation (2.28)
can be less than zero the possibUity exists that SNRoj could equal zero for nonzero a. The
most interesting behavior is displayed by SNR02. For very small a, SNR02 is near the
maximum value SNRo2= NSNRi2. As a increases away from zero, the SNR02 function rolls
off at the -3 dB point of a^ =p2b and continues to decrease until a^ = Z2b ( SNRii =
[Nsin^ai2\-^ )• At a^ = Z2b the SNR02 roll-off attenuation is halted and SNR02 begins
increasing since die pole z.ip2b is canceled by the zero at a^ = Z2b. Over the range of a
specified by this scenario, the point a^ = Z2b represents a minimum of SNR02. Substituting
a2 = Z2b into equation (2.28) gives SNRoj = 1 and SNRo^ = 4NSNRiJ{NSNRi^ + 5) - 4 .
36
Note that similar expressions for the case of orthogonal users ( case a above) can be
readily found by simply setting sin2ai2 = 1 and cos2ai2 = into equation (2.28).
In summary, for most of scenario 2 the array is unable to resolve the low-power user 1
input signal from the input noise. The array can just begin to resolve the user 1 input signal
from die input noise when a^ = Z2b ( SNRii = [//sin2ai2]-' )• At this point, SNRoi = 1 and
SNR02 = 4 which represents a minimum value of SNR02 over this scenario.
Two Equal-Power Users
If the two users have equal input power levels die output SNRs are equal:
f^^f^Ri (1 + cosa,2 + NSNRi sin' a,, Y
SNRo = ^^ —
2 1 + cosai2 + NSNRi sin' a,^ \2 + NSNRi{\ - cosa,^ ))
(2.30)
and for die case of well-separated users (case b: < cos2ai2 « sin2ai2 < 1) die output
SNRs may be approximated as:
NSNRi (NSNRi sin^ a,^f
SNRo ==
(2.31)
2 NSNRi sin' a,^ (NSNRi{l - cosa^^ ))
NSNRi I X
= -^— (l-hcosajj
If die users are spatially orthogonal {case a) dien sin2ai2 = 1 and cos2ai2 = and die
output SNRs are exacdy equal to SNRo = NSNRi/2, which is consistent widi equation
(2.6).
Two Strong ILsers
For diis scenario (#4) we have two strong users widi unequal input power levels.
For case b ( < cos2ai2 « sin2ai2 < 1 ) die numerator terms of equation (2.26)
may be modified by multiplying all SNRi2 terms by an additional sin2ai2 tenn. The
37
denominator (equation (2.23))changes slightly:
A, =aVa^
+ 1
N' SNRi; sin' a ^-, NSNRL, J N^SNRi^&in^a^^
(2.32)
Terms containing higher powers of a dominate the numerators (equation (2.22)) and
denominator (equation (2.32)) of the output SNR expression (2.24). This condition
represents the presence of two sti-ong users of unequal power: SNRoj » SNR02. The
expressions for the (scenario 4, case b) output SNRs may be approximated as:
SNRo. = ^^ ^— — i^
NSNRi^sin^a^^
SNRo, --NSNRLsm^a
(2.33)
12
Note tiiat for large input SNR the expressions in equation (2.33) should be nearly equal.
The difference between die output SNRs for tiiis scenario is:
ASNRo = SNRo, - SNRo^ = 2 +
NSNRi^sin^a^^
= 2 + -
1
SNRo.
(2.34)
Which is very similar to the expression for output spread for orthogonal users given by
equation (2.14).
Numerical Results
This section will first present numerical solutions to corroborate the two-user analysis
of die previous section. The data will be presented in plots of the output SNRs (equation
(2.26)) versus Uie input near-far ratio (a^). Array performance will be evaluated for
spatially orthogonal users and well separated users evaluated over a wide range of a^.
38 : " ,
Specific comparisons between analytical and numerical solutions will be made for critical
points identified in the analysis.
Performance for a linear MMSE array was investigated by plotting the output SNRs
of two signals while varying one signal input level and holding the other at a fixed value of
input SNR. For each point the MMSE weights and corresponding output SNRs were
calculated using equations (1.6)-(1.10) and equation (2.1). The DOAs were fixed to
specified values. The plot below shows the output SNRs for the case of a three element
array with two users. User 1 is positioned broadside to the array (DOA = degrees) with
varying input SNR, and user 2 is positioned at 41.8 degrees from broadside with an input
SNR of 10. This DOA condition gives the least amount of source separation which allows
the sources to be spatially orthogonal.
Figure 2.2 shows the (A^ = 3) array output SNRs versus a' for a two-signal scenario
over the range 10-5 < a^ <\^. The users are spatially orthogonal, ai2 = 90 degrees and
the user 2 input SNR is 10. For a^ = 10-5 the array is in essence a single user system.
SNR02 = mNRi2 = 30 which is die maximum possible output SNR for user 2 and SNRoi
= 0. As a {SNRii) increases, SNR02 begins its roll-off attenuation: SNR02 rolls off 3 dB by
the time a^ = .001 1, as predicted by equation (2.28). As a increases the SNR02 roU-off
continues until it reaches the minimum value of 3.33 at a^ = 0.033. At this point the array
is just able to resolve signal 1 from the input noise and SNRoi = 0.89. These numbers
agree well with equations (2.10) and (2.28).
39
Figure 2.2 Output SNRs versus near-far ratio for spatially orthgonal incident
signals
For the scenario a^ = i the output SNRs are equal to 15 (see eqn. (2.6)). As a^
becomes very large the output SNRs lose their dependence on a^ and approach nearly the
same value. This is shown in equations (2.33) and (2.34). At large near-far ratios (a^ =
10 ) SNRoi = 32.03 and SNR02 = 30. The difference between their values is ASNRo = 2.03
which agrees precisely with equations (2.14) and (2.34).
Figure 2.3 shows the performance for DOA spacings greater than the Rayleigh limit
For this scenario user 1 remains at array broadside and the DOA for user 2 moves to 90
degrees (in line with the array). This case gives the vector quantities VjV2= Ncosai2= -
\, cos^ai2 = 1/9 and sin2aj2 = 8/9. When a^ is smaU (a^ = 10"') SNR02 = NSNRi2 = 30
as in the previous case. As a^ increases to 1.18x10"^ SNR02 rolls off approximately -3 dB
40
to a value of 15 and SNRo] increases from nearly zero to 6.5x10 ^ Note that equation
(2.28) approximates SNRoj = 5x10'' and a -3 dB roll-off point of a^ = 1.25x10 ^
At a2 = 0.042, SNRo, = 0.798 and SNR02 takes on a minimum value of 2.807.
Equation (2.28) predicts a minimum value of SNRo2= 3.87 will occur for a^ = zib =
0.0375 and SNRo, = 1.01. From the plot SNRo, = SNR02 = 9.94 when a^ = 1 which
agrees with the approximations given by equation (2.31). When a^ increases beyond unity
the outputs quickly level to values near the maximum of SNR02: SNRo, = 28.64 and
SNR02 = 26.61 at a^ = 1x10*. This gives an output spread (eqn. (2.34)) of ASNRo = 2.03.
Figure 2.4 shows what happens when the DOA spacings are less than the Rayleigh
limit In this case, signal source 1 remains at broadside while source 2 moves to DOA2 =
Figure 2.3 Output SNRs versus near-far ratio for nonorthogonal incident signals
with DOA spacings greater than the Rayleigh limit
41
20 deg., about half the Rayleigh limit. The outputs are degraded for a^ » 1, compared to
the two previous plots.
Output
SNR
10^ Near-Far Ratio: a^ 10*
Figure 2.4 Output SNR versus near-far ratio for incident nonorthogonal signals
with DOA spacings less than the Rayleigh limit
CHAPTERS
THE MULTIUSER ARRAY PROCESSOR
ADAPTIVE PERFORMANCE
Consider for a moment that filter weights in an adaptive process are random variables
with salient statistical properties. Adaptive processors are unable to perfectly track the
corresponding steady-state solution; this induces a penalty known as misadjustment'^ The
goal of this chapter is determine the effect of the step size of the complex LMS
algorithm on the steady-state SNR-leveling performance of the multi-user processor.
This concern is motivated by two factors. First, from an intuitive standpoint, it seems that
the multi-user processor might cease to level the output SNRs if the misadjustment
becomes too large. Second, the multi-user array processor might share some similarities to
an LMS automatic line enhancer (ALE) for multiple time signals. Fisher and Bershad"
studied the misadjustment performance of an LMS-ALE for the case of multiple sinusoids
in AWGN and found the equaUzer misadjustment to be especially sensitive to the step size.
This work will rely heavily of the work of Senne"^ who developed expressions for the
time-dependent and steady-state weight covariance matrix of a real LMS adaptive
processor. Other authors have investigated the transient and steady-state behavior of the
complex LMS weight covariance matrix.^^'^^ and have found that the steady-state
eigenvalues are very similar to those derived by Senne for the real LMS algorithm. This
detail will be applied in a later section of this chapter.
42
43
Other authors have examined the performance of adaptive processors subjected to
hard and soft constraints on SNR as well as optimizations of SNR itself subject to
nonlinear constraints.'*''
This chapter will be divided into three sections. The first section will give the
development of a performance measure (cost function) which is based on the mean output
SNR of the adaptive array. The second section develops a performance measure based on
the variance of the array output SNR. The last section presents numerical results which
allow comparisons between the analysis and simulations. The cost functions will be used
to find acceptable bounds on the LMS step size. It is found that the upper bound on step-
size arising from stability arguments also gives acceptable output SNR performance for
the multi-user processor.
Mean-Based Performance Measure
If the adaptive weights of an LMS processor are considered to be time- varying
random variables, w(n) = Wo -h v(n;, the mth user's time-dependent output SNR results
from substituting the expression for w(n) into equation (1.10):
|(Wo+v(/i))'"u
SNRo{n)„=SNRi„
(3.1)
The random, time-varying weight component is represented by v(n), the mean of the
weight vector is the steady-state MMSE weight vector, Wo = R"^p, and the discrete time-
dependence is introduced by the variable n. Note that the phase-shift vector u„ and SNRi„
are constants.
44
In order to derive an expression for the performance measure we need to find the
mean of the output SNR. Some assumptions facilitate this effort:
1. ||v(n) + w„|| = ||w„|| ; instantaneous deviations of norm(w) from
norm ( Wo) are negligible because of small step size ^i.
2. v(/i) are independent from sample-to-sample.
3. Input signals and noise are i.i.d., stationary and ergodic.
4. The elements of w(n) form a jointly Gaussian process.
The resulting mean of the output SNR is:
E{SNRo{n)^ ] = SNRi„ ■ ^^ + SNRi„ ■ ";^-"" (3.2)
W W
where the first term on the right-hand side is SNRom , the steady-state output SNR defined
earlier in equation (1.10). The matrix Cw = cov(v(n)) is the steady-state weight covariance
matrix given by Senne.'' The matrix is not a function of time, n, since v(n) components are
i.i.d.
The last term on the right-hand side will cause the average output SNR for user m to
increase as the adaptive weight covariance increases. This might have Utde effect for weak
users; the term is not negligible for strong users. The average output SNRs may no longer
be leveled if the term becomes too large. This point is clarified in Appendix A which
presents some general characteristics of the SNR performance surface.
The first measure of performance is the difference between the average and steady-
state output SNRs divided by the steady-state output SNR:
E{SNRo{n)„]-SNRo„ _ SNRi^ nlC^y„
SNRo„ SNRo„ ||w„f - • ^^-^^
Where SNRom is the steady-state output SNR for user m and is defined by the first term on
45
the right side of equation (3.2) above. The quantity Ci »= l/(NSNRi2) « 1 is a constant
which serves as an upper bound which will limit the excursions of the time- dependent
SNR (resulting from the adaptive process) from the steady-state solution. The constant
will be defined in more detail later in the analysis. If the adaptive process is allowed to
stray too far from the steady-state solution the output SNRs may no longer be held to
nearly the same level and near/far limited performance may result. Using this relationship
with equation (3.2) above gives an expression in which step size dependence is expressed
indirectly through Cw:
T^ • ^ /^ II l|2 S^^O„
u«C„u„ < C, wj (3 4)
SNRi^
The quantity on the left-hand side may also be lower-bounded by using the maximin
theoreom^^:
mm
mm
'r(Aj = qeC-
q?tO
V
(3.5)
q q J
where C^ denotes the complex A^-dimensional space, q is an A^-length vector and ?lw =
minfaU eigenvalues of €„}. Let q of equation (3.5) equal u„ of equation (3.4) where
Um Utn*=N. Equation (3.5) may then be used to form a lower bound for equation (3.4).
The left-hand term of equation (3.4) would then be bounded by:
'V-'^V)^"IC„u:.C,|kr^ (3.6)
The next task is to find a useful expression for the lower bound of equation (3.6).
Senne"^ has shown that the weight covariance matrix is diagonalized by the eigenvector
matrix of R, the input autocorrelation matrix. For an ^ = 3 array, the diagonal of Av^, the
46
eigenvalue matrix of Cw, is:
diag{A„} =
^K
1-1
2
( M,
U-M
ly
M2 "I
l-;iA
2 J
' M3 ^
U-M
3/
(3.7)
where A;, A2, and X3 are the eigenvalues of R, and |i is the LMS step size. Compton'"*' has
shown that when SNRi] » SNRi2 > 1, the signal and noise eigenvalues of an N-by-N
autocorrelation matrix R are approximated by:
A, =(T^(l + A^-5A7?i,)
f (
K-o]
K=ol
l+SNRL
V
N-
I T '\^W
N
(3.8)
yj
m = 3...N
where the largest eigenvalue A/ is established by the strongest incident signal and A™ are
the noise-only eigenvalues.
If the step size is small (ji < 7/A;)the smallest Aw from equation (3.7) may be
approximated by:
"'"{^4=-^(i-M,)
1--
/iA,
If ^X,
1-//A,
if nX,
1-//AJ 2ll-/iA
3y
(3.9)
;^e:
The simplification m equation (3.8) results directly from our assumption of small step size.
If jj, is small the term in brackets and the term in parentheses outside of the brackets are
equivalent to unity.
47
This establishes the upper and lower bounds on u'^C u*
m vv m
^mel,^ < uXX < C,Kf ^^ (3.10)
^ SNRi„
If all the incident signals are well-separated, strong and perfectly correlated to the
reference waveforms the minimum MSE should be well-approximated by the output noise
power: e^, = al ||wj . If user m = 1 is the strongest user then SNRii can be determined
in terms of the eigenvalues via equation (3.8) (N- SMRi^ = (A,/<t^) - 1 = A,/ct^ ). The
constant Ci from equation (3.3) establishes a bound on the average excursions from the
steady-state output SNR due to misadjusttnent, so it must be made suitably small. The
strongest user - user 1- should have a steady-state output SNR leveled to approximately
the same value as user 2, therfore SNRoi = SNR02 = NSNRi2. Since even SNRii » 1
then Cj = nJO-NSNRii) should give acceptable results where |io is a constant (0 < //<, < 1)
and represents die unormalized step size . This results in an upper bound on //:
which is similar to the upper bound on the LMS step size derived from stability
22 93 J •
arguments, • and is consistent with our earlier assumption of a small step size.
Variance-Based Pe rformance Mea.snr^.
The second cost function is the variance of the adaptive output SNR divided by the
squared mean of the adaptive output SNR. Choosing \i to keep this measure small will
limit excursions from the mean of the adaptive process. The second moment of output
SNR is:
48
E[SNRoin)i] = ^-d{y,„+y{n)fuS] (3.12)
Senne assumed that the filter weights, w(n) = Wo + v(n), form a jointly Gaussian process.
In the following analysis this assumption is used to expand fourth-order joint moments in
terms of the second-order moments via the Gaussian moment factoring theorem.'"'
If we assume the vector v(n) is multivariate and complex Gaussian we can express the
fourth order moments as functions of the second-order moments.*"' Expanding the
bracketed vector term and discarding terms that would give odd-ordered moments of v(n)
results in several terms:
4|K + v)'u„p] = |wXf + 4 • |w>„|'<C„u:
+ 4wX)»:)%(wX)»„f] (3.13)
N N \ S
where we have dropped the time-dependent notation from the weights. The third term on
the right is zero. The fourth-order moment may be expanded in terms of the second-order
moments:
4^ K ^ K ] = 4^'. < ]4^ K ] + 4^ ^i ]4^ ^ ] (3. 14)
The simimation term becomes:
IIII"',<"/,"/:4^^^K]=2-Kc,,u:)^ (3.15)
^-1 1,-1 1,-1 1^=1
The second moment of the output SNR is:
'^011
(3.16)
49
and the variance of the output SNR is (from equation (3.2)):
|wXf+u:C„u:)'-|wXf
f^SNRoJin) - SNRi„ II ||4
w
oil
>l
= E[SNRoMY - SNRol
where the second line of equation (3.17) follows from (3.2) and SNRo„ is the steady-state
output SNR of the m'th signal. The performance measure becomes:
^SNRo(n)_ , SNRoi
= 1-3 TT-^- ^ C2 (3.18)
45A^^o(n)J £[5A^^o(az)J
where C2 « l/(NSNRi2) « 1 is a constant to be defined later.
Substituting quantities from equation (3.2) into equation (3.18) results in the
inequality:
SNRo
u^C u* < llw
m ^ vv m
VT^Q
' -1
(3.19)
Since C2 < 1 the radical term may be approximated by a two-term series
(1 - C2) 2 = 1 + C2/2 and the inequaUty takes the same form as equation (3.4). The same
procedure used to determine the step size upper bound for the first performance measure
may be followed for this second performance measure with the same results: /i < ^JXi .
Simulation Result.s
This section will use simulations to corroborate the steady-state analytical results of
previous sections.
so
The next two plots show curves of the quantities obtained from equations (3.2) and
(3.18). Figure 3.1 shows the average output SNRs versus the step size coefficient for
means resulting from simulation and from expression (3.2). The input scenario is SNRii=
10, SNRii = 10^ DOA] = deg., DOA2 = 90 deg., A^ = 3. The initial weight for the LMS
simulation was MMSE weight vector from equation (1.8), Wo = R'^p.
Plots of the variance-based performance measure are shown in Figure 3.2. The input
scenario is identical to the one described in Figure 3.1. The curves show the ratio of
variance divided by the squared mean of the output SNR obtained tiirough analysis
40
35
Average output SNR vs. step-size coefficient
m
3 30
o
I
25
20
I I I ip
SNRi1 = 10 dB
SNRi2 = 60 dB
D0A1 = deg
D0A2 = 90 deg
User 2: sim
I I Ill ll-H
User 2: calc
I M III III III II II II III I n il II II II II II lllll II III
User 1: sim, calc
10^
10-* 10-'
Step-size coefficient
10"
Figure 3.1 Output SNR versus step-size coefficient |io. The step size is
normalized by the input power
(equation (3.18)) and simulation. The curves show, as we might expect, that the
performance will be worst for the strongest user. Selecting step size for acceptable strong
user performance will lower-bound the performance for the remaining weaker users.
51
10'
2nd moment perfomiance measure vs. step size coeff cient
10"
lO"
10"
Step size coefficient
10"
Figure 3.2 Variance-based estimator versus step-size coefficient (Xo.
The step size is normalized by the input power. The input scenario is
identical to the one described in Figure 3.1
Selecting a small performance measure will limit the point-by-point excursions of the
output SNR from the steady-state value and will give results which are acceptable in the
mean. If 1% is selected as an acceptable value of the performance measure then from the
curves for user 2 above it would appear that ^ = 0. l/Tr(R) is the maximum allowable step
size. This is in agreement with existing rules-of-thumb for selecting step size for stability
and convergence. For these conditions, it would appear that upper bounds on LMS step
size derived from convergence arguments will give acceptable performance of the
multiuser processor.
CHAPTER 4
BASE STATION RECEIVER PERFORMANCE USING A SMART ANTENNA
ARRAY IN A DS-CDMA SYSTEM WITH IMPERFECT POWER CONTROL
This chapter examines the steady-state outage probability and outage-based capacity
of a single cell containing multiple directional signal sources transmitting to a central base
station. The only fluctuation in the signal levels is due to lognormally-distributed power
control error in the multiple transmitters. The central receiver consists of an array of N
isotropic sensors, K minimum mean-squared error single-user beamforming processors
and a bank of conventional detectors. The performance measures are outage probability
and outage-based capacity. The goal is to find simple expressions which relate the outage-
based capacity to the antenna array parameters, the number of active signal sources and
the degree of power control error. The analytical results are expressed in terms of the
number of users per array element which may be supported for a given outage probability.
Analytical results are found to agree closely with those obtained from Monte Carlo
simulations.
This work is unique in several respects. First, most previous work examined uplink
performance for a traditional single-channel receiver. Second, the authors which
considered a base station antenna and imperfect power control presented simulation
results for the case of closed-loop power control.*^*^ As was mentioned in Chapter 1,
power control error will be modeled by incident power levels which are lognormally
distributed. This model has gained some acceptance for open-loop power-controlled
52
53
systems. Its suitability as a model for closed-loop power-controlled systems remains an
open issue.
The remainder of the chapter is divided into several sections. A qualitative description
of the array and the receiver is given in the next section. The third section outlines the
development of the analytical model and the derivation of the expressions for uplink
capacity. The fourth section describes the Monte Carlo simulations and compares the
analytical and the simulated results. The last section presents the conclusions.
Svstem Description
The incident DS-CDMA signals are independent and originate from independent
transmitters which are arbitrarily placed about the base station receiving antenna. The K
active transmitters (located in a single cell) result in incident signals with independent
directions-of-arrival (DOA) uniformly distributed over the interval [0, lit). The
transmitters' output levels are continuously adjustable and have an infinite dynamic range.
The output power - adjusted by an unspecified power control algorithm - results in output
power which is independent between sources and which is also lognormally distributed.
The modulation format is BPSK.
The base station receiving array consists of A^ ideal isotropic sensors arranged in a
circle. Adjacent sensors are separated by a distance with the electrical equivalent of one-
half wavelength. The sensor array feeds K separate banks of ^V complex weights
controlled by K MMSE beamforming processors. The optimum steady-state weight
vector for each beamformer is given by equation (1.8), Wo = R'^p. Each of the K
beamformer outputs feeds a distinct conventional DS-SS detector which in turn provides
54
an estimate of the demodulated output. The receiver is assumed to be capable of perfect
carrier tracking of the desired signal. Unlike the multiple-user array of the previous
chapters, the array/detector combination is devoted to detection of a single desired signal
amidst multi-access interference. A diagram is shown in Figure 4.1.
Conventional
DetectOT 1
bat,]
Conventional
DetectOT 2
Conventional
DetectOT K
I'ea.K
Figure 4.1 A bank of baseband single-user beamformers and conventional detectors
sharing an array of sensors
Analysis
The stationary, complex base-band output of the sensor array is found by combining
equations (1.3) and (1.4):
55
m = At,iO''"'c„it-r„)b„{t-T„)u^+n{t) (4.1)
msl
where A is the peak amplitude and €„ is N(0,apc/). The sensor outputs are weighted and
summed by the beamformer and then processed by the conventional detector which forms
an estimate of the current output bit. The signal from source 1 is considered the desired
signal. The remaining K-1 signals are considered multi-access interference.
The array tends to steer a pattern lobe towards the desired signal and also tends to
steer nulls of finite depth towards the N-2 strongest interferers when K >N-L Because
the strongest signals are attenuated, no single signal dominates the output statistics and the
output may be well-approximated as Gaussian. An additional assumption of long codes
(codes which span more than one data symbol) allows the PN sequences to be modeled as
random codes. The random code model of Pursley'°^ will be used here. More refined
models exist, but the additional complexity they introduce tends to improve accuracy
when only a few users are present. '°3-'**-io5
From Pickholtz et al.'°* we know diat the effective bit energy to noise spectral density
ratio may be approximated by (Eb/No)eff^NcSINR. The random code approximation may
be modified to account for varying input amplitudes among the incident signals due to
their power control error. Using one of the intermediate steps (equation (14) from
Pursley' ) allows a convenient expression for {Eb/No)eff while retaining the cross-
correlative properties of the PN codes:
^ E ^
X, c-rxrr, N.SNRo,
= N^SINR = — — j^ ! (4.2)
c m=2
where SNRo„ is the array output SNR of the mth incident signal and r;,^; is a
36
crosscorrelation parameter between the first and mth PN codes. If SNRom is replaced by its
sample mean and the remaining sum of r„^i terms is replaced by the random code
approximation (equation (16) from Pursley'"^) the expression becomes:
El
N.
NSNRo,
= Y^ ■ (4.3)
^ m=2
The quantity SNRok may be rewritten as:
SNRo„=\Q''"'.Gp„SNRi (4.4)
where Gp„ = |wju„| /||wj| is the normalized power gain of the array towards the mth
signal and SNRi = A^/On^.
The power gain is a complicated function of the input scenario and the array
geometry. In order to simplify the analysis we will use a simplifying assumption that the
array power gain with respect to the first user, Gp, = Gpa, wiU be approximately equal to
its upper bound A^, the number of array elements. The MMSE array gain towards the
interferers (Gp„,m = 2 ... A) is difficult to characterize. Intuitively, we might expect that
the avg{ Gp„ } = Gpi might be well-approximated by the average (over DO A) normalized
power gain (Gpayg) of a classical beamformer (w, = uj with no adaptive null-steering
capability. The value of Gpa^g ranges from unity for a single element to 1.6 for a 30-
element circular airay. Simulated results presented shortly will show that these
assumptions with regard to the individual quantities (Gp^, Gpd are not always valid over
the conditions of interest. The average of die ratio GpJGpi .however, does provide a
reasonably good fit to die simulated results when die above assumptions (Gpd - M Gpi «
N) are used.
57
If there are many interfering signals the sum in the denominator of equation (4.3) may
be approximated by averages:
r T7 \
E,
N„
NMSNRilO
x/\0
^w-, (4.5)
V"" y^ ^K - l)SNRiGpilO ^oo ""' + 1
Further explanation is required. The sum in the denominator of equation (4.3) is almost a
sample mean and may be approximated by an ensemble average. Assume the power gain
and the exponential of equation (4.4) are independent: the expectations apply separately.
This results in the base- 10 exponential term, the (K-1) term and Gpi in the denominator of
(4.5). This approximation causes a slight overestimation of interference: by breaking a sum
of products into a product of sums we have invoked Schwartz's inequality twice in
succession to get from (4.2) to (4.3) and to get from (4.3) to (4.5). Our simplifications
have equated sample means and ensemble averages and also ignored the complicated
nature of the array response by treating the power gain as an averaged quantity.
Simulations of the power gain show our assumptions regarding Gp may give
acceptable results. Figure 4.2 shows the results of Monte Carlo simulations of averaged
values of (MMSE) Gp for a desired signal (Gp^), an interferer (Gp, ) and their quotient for
a varying number of users, 3 dB of power control error and a 30-element array. The
curves show that as the number of users exceeds the degrees-of-freedora of the array our
simulated values for the individual gains are not very close to approximated values of Gpd
= iV = 30 and G/?„ = Gpavg = 1.6 but their quotient GpJGpi, = A^ = 30. So, for this
circumstance it might be useful to approximate the gains in equation (4.5) as Gpd = A^ and
Gpi, = 1.
58
Why will this work? As the number of users increases and the system becomes
interference-limited the "1" term in the denominator of equation (4.5) becomes negligible
leaving a good approximation of the quotient of the gains as Gpd/Gp, ~ N. When there are
few users the multi-access interference is negligible, Gpd ~ N and the approximation will
still hold. These gain approximations do not hold separately to predict the desired signal
output power or the interference output power, but their combination might prove useful
in calculating outage.
What about more extreme scenarios? Figure 4.3 shows the gains and the gain ratio for
a power control error of 10 dB. The individual approximations Gpd « A^ and Gp„ = Gpavg
= 1.6 are even worse than before. However, the simulated gain ratio Gpd/Gpi = N/L3 for
K>N. This is a little closer to the ratio of the individual gains Gpd /Gpavg . = N/1.6. This,
1Cf
10"
Gain
ICP
ia'
1 1 1
'
^/^ — ^— — ^
Gp/Gpi
^""^^^
■
Op4
■
Gpi
power control error = 3 dB
1 1 1
N = 30
L. 1
20 40 60 80
Number of Incident Signals
100
120
Figure 4.2 Gain versus the number of incident signals. Power conu-ol error is
3dB.
59
^(f
QP..t:.K=M.
Gain
^(f
^Q■'
power control error = 10 dB
N = 30
20 40 60 80
Number of Active Users
100
120
Figure 4.3 Gain versus the number of incident signals. Power control error is
10 dB.
and other simulations, indicate that choosing Gpd = A^ and Gp, = Gp^v^ = 1.3 for this
model is a good approximation for outage calculations over the ranges of power control
error examined here (0 < Opec < 10). Note that Gp, = Gpa., = 1.3 also arises from a
sample mean (over A9 of the average power gain (over DOA) for A^= 1,2,4,8,15 and 30-
element arrays for a classical beamformer.
Approximating the interference as an average quantity in equation (4.5) eUminates
complex scenario-by-scenario interactions between the desired signal and the interference
in the analytical model. It also leaves the lognormally distributed desired signal as the only
random variable in the model. The resulting distribution of the (Eb/No)eff in equation (4.5)
is therefore log-normal and results in simple expressions for the outage probability and the
60
capacity. Outage occurs when the {Eb/No)es is less than some threshold. Converting
equation (4.5) to dB and noting that ei is N(P,<J^pcc) results in a simple expression for the
outage probability:
PTou,,,. =Pr
W^" ^eff.dB
< ^
dB
= Q
ZdB-^
dB
(4.6)
V /« /
where Q(x) is the complementary Gaussian CDF, ^b is the desired threshold in dB and:
Z,fl = 10*log
10
k
D
J
(4.7)
The quantity D is the denominator of equation (4.5). The quantity Eb/No « (Tb/Tc)SNRi
=NcSNRi represents the equivalent bit energy to noise density ratio for a single incident
signal and a single array element. Note that Naguib et al.^* has also developed a Q-
function upper bound for outage probability for the case of perfect power control. For the
case of imperfect closed-loop power control Naguib et al.*' presented the simulated means
and variances of (Eb/No)eff.
Some simple algebraic manipulations result in the average capacity as a function of
desired outage probability:
K = C
where:
f'«
i — ^
(pf~».)
v-l
N.
oj
+1
(4.8)
IGp,
where for high Et/No the capacity is approximately linear in processing gain, Nc, or the
61
number of array elements N but decays exponentially as the power control error increases.
Note also that the threshold ^ is no longer in dB. Note that if we interpret equation (4.8)
as being linear in N, we can define Si per-element capacity by noting the slope of the line.
Note also that the capacity asymptotically approaches a finite maximum as Eh/No
increases. A similar effect was noted by Naguib and Paulraj*^ for the case of a 2-D RAKE
combiner at the base station receiver and perfect power control in the mobiles. They
examined the capacity as the equal incident signal levels went to infinity and named the
parameter asymptotic capacity. A simple expression for uplink capacity was also
formulated by Suard et. al.'^ for a post-detection combiner. The model for power control
error was restricted to a single user with an incident power level 10 dB higher than the
other users.
Simulations and Results
Monte-Carlo simulations were used to corroborate the analytical results given by
equation (4.6). Autocorrelation matrices for the desired signal, interferers and noise were
generated by ensemble averages as in equation (1.6). Signal DOAs were uniformly
distributed over [0, 2ic). The processing gain was 127 and Eb/No = 7 for a single antenna
element and no power control error. For a single trial the MMSE weights were calculated
via equation (1.8) and the desired signal, interference and noise power out of the array
were then used to calculate (Eb/No)eff via equation (4.3). An outage condition was judged
to exist for that trial if (Eb/No)eff <7dB (i.e. from equation (4.6), ^b=7 dB). The
quantities were averaged over 20,000 trials for each combination of power control error,
number of users and number of array elements.
62
Curves of the outage probability - ?x(SEb/No)eff < 7 dB) - versus the number of users
are shown in Figure 4.4. The figure contains curves from equation (4.6) as well as the
Monte Carlo simulations. Power control error is 4 dB. Note that for 20,000 trials, curves
in Fig. 4.4 might be inaccurate for outage less than 10"^.
Figure 4.5 shows the total array/receiver capacity versus the number of array elements
with outage probability as a parameter. The curves are formed by plotting constant
contours of a three-dimensional surface formed by ?T(,(Eb/No)eff < 7 dB) as it varies over
K and A^. The solid curves show simulated results; dashed lines show the constant- value
contours of equation (4.6) for power control error equal to 4 dB.
The curves of Figure 4.5 show the user capacity of the array/detector is roughly linear
in N. We may therefore use as a performance measure per-element capacity, the number
of users per array element which may be supported for the given values of outage
probability and power control error. As noted earlier the analytical expression for the per-
element capacity may be obtained by noting the slope of the total capacity line in equation
(4.8) with A^ as the independent variable. A point of note: the average power gain {Gpavg)
in equation (4.8) has a slight dependence on A*^: it is equal to unity for a single element and
is equal to 1.6 for a 30-element array. For the sake of simplicity, this slight dependence is
ignored and a mean value of Gpavg = 1.3 is assumed, which was noted earlier when
comparing simulated Gpd/Gpi curves in figures 4.2 and 4.3.
The per-element capacity versus the power control error for Pri(Eb/No)^ < 7 dB) =
0.02 is shown in the upper plot of Figure 4.6. For simulated data, the normalized capacity
was determined by extracting the approximate slopes of die capacity curves (as shown in
Figure 4.4) via a linear least-squares curve fit. The analytical results were calculated from
63
Outage
Prob.
N=15
N=30
Simulation
Analysis, eq. (8)
20 40 60 80 100 120 140
Number of Incident Signals
Figure 4.4 Outage probability versus the number of incident signals. Power
control error is 4 dB. The number of array elements N is a parameter
80
Capacity
70
10 15 20
Number of Array Elements
Figure 4.5 Capacity versus the number of array elements with outage probability
as a parameter. The power control error is 3 dB.
64
Per 30
Element
Capacity
20
10-
Intercept
-10h
-20
-30
-40
8 10
Power Control Errw (dB)
1 1 1
• ■ ■««
........... I .......... J .... -^n.. .
— '"/t
, , - - -J.—-^- - • -
^ — -r
/
__ L„^-^' ;
■
^Vy; i
•
•
Simulations
- Analysis, eq. (4.8)
(b)
8 10
Power Control Error (dB)
Figure 4.6 Normalized array capacity versus power control error in dB.
the slope of the system capacity line given in equation (4.8). As the curves show, the
analytical and simulated results for the per-element capacity are in close agreement. The
lower plot of Figure 4.6 shows the intercept point of the line in equation (4.8). Obviously
the analytical and simulated results are not in as close agreement as the upper plot. Note
that to determine the overall system capacity - given an array of iV elements - it would be
necessary to know the per-element capacity as well as the intercept
65
Conclusions
In this chapter we attempted to develop accurate, simple analytical expressions for the
outage-based uplink capacity in an idealized single-cell DS-CDMA system with multiple,
possibly near/far, signals incident on a base-station receiving array. Power control error
was assumed to be lognormally distributed. Some simplifying assumptions regarding the
interference and the array response allowed an approximate expression for the uplink
capacity that was compared with results from Monte Carlo simulations.
The approximate expression - given in equation (4.8) - shows that a roughly linear
relationship exists between the capacity and the number of array elements. The overall
capacity K consists of two components. The first component is the slope of the line in N
and has been defined as the per-element capacity, the number of users per array element
which may be supported for a given level of outage probability and power control error.
Equation (4.8) indicates that the per-element capacity is not dependent on the nominal
input level (i.e. the input level without power control error: SNRi) but decreases
exponentially with increasing power control error. For the levels of power control error
examined here the term which dominates the exponential roll-off is (JpceQ''(Prou,age)/W of
equation (4.8). The decrease in the per-element capacity is therefore dependent on the
outage requirements and the standard deviation - in dB - of the power control error. Since
the per-element capacity is insensitive to the nominal input levels and the array size, but is
keenly dependent on the degree of power control error and the outage, it might serve as a
useful asymptotic measure of the performance for this array/receiver.
66
The second component of equation (4.8) is the intercept term equal to l-C(Eb/No)''
where C is defined in (4.8). The intercept is inversely proportional to the nominal input
levels. As the input levels decrease this term becomes larger, diminishing the overall
capacity. In a plot of capacity (K) versus array elements (N), the capacity line moves away
from the origin along the horizontal N^-axis as the nominal input levels decrease. The
intercept is weakly dependent on the power control error (via Q and also on the number
of array elements (via Gpavg) and is not dependent on the outage probability. Analytical
and simulated results do not agree as closely as those for per-element capacity.
The agreement - or lack of it - between the analytical and simulated results for per-
element capacity and the intercept can be interpreted in terms of outage probability curves
shown in Figure (4.4). The analytical model can predict well the horizontal spacing
between the continuous outage probability curves. The per element capacity predicts the
incremental increase in capacity with increasing array elements and is a measure of the
horizontal displacements of the outage curves relative to one-another. The less reliable
prediction made by the model is the horizontal placement of the outage curves relative to a
point on the horizontal axis. This enters into the model via the intercept parameter
described above.
The simple model presented here was based on some simphfying assumptions
regarding the array response towards the incident signals. In spite of this, the analytical
model accurately predicts some aspects of the array/receiver performance when directional
signals are employed with lognormally distributed power conttol error. The directional
signals originated in a single cell and were not subjected to any kind of environmentally-
67
induced fading. The system performance in the presence of fading and interference
resulting from outer cells is the topic of the next chapter.
CHAPTERS
BASE STATION RECEIVER PERFORMANCE WITH A SMART ANTENNA
ARRAY IN THE PRESENCE OF MULTIPATH FADING AND SHADOW FADING
This chapter examines base station receiver performance when incident signals are
subjected to fnequency-nonselective Rayleigh multipath fading and lognormally distributed
shadow fading. The effects of multi-access interference from outer cells - subjected to
shadow fading - wUl also be included in the incident signal model. The power control
error model will continue to be described by a lognormally-disttibuted random variable
and the receiver still consists of an array of A^ ideal isotropic sensors, K minimum mean-
squared error single-user array processors and a bank of conventional detectors (see
Figure 4.1).
The goal of this chapter is to determine the performance dependence on the number
of active signal sources, the number of array elements and the degree of power control
error. As before, the performance will be expressed by outage probability and per-element
capacity. Unfortunately the introduction of fading and outer cell interference further
complicates the development of simple analytical models. In spite of these complications
the simulated results closely follow some general trends estabUshed by the model in
Chapter 4. In particular, curves of outage-based capacity continue to be linear in A^ with
slopes that decrease exponentially with increasing power control error.
Previous authors have investigated the performance of optimum combining from the
standpoint of interference rejection. Winters"'" and other authors^'^' have examined
contributions to TDMA system performance while Naguib et al.^^** have studied CDMA
68
69
systems. Unlike previous work, this research examines the outage-based capacity for the
case of imperfect power control and attempts to provide some analytical models which
would allow easy assessment of performance.
The remainder of this chapter is divided into six sections. The first section will present
the receiver model for single-cell signals subjected to Rayleigh fading. The second section
will extend the model to the multi-cell case. The third section will quickly revisit the model
of chapter four and introduce a second analytical technique. The fourth section will give a
brief description of the simulation parameters and compare the simulated results with the
analytical results.. The fifth section entitled Conclusions and Discussion will review the
results in some detail. The last section gives a summary.
A Single Cell with Signals Subjected to Ravleigh Fading
Multipath fading arises when propagating electromagnetic waves originating from a
single signal source arrive at a receiver via different propagation paths. The individual
paths may include Une-of-sight propagation as well as paths resulting from reflection off of
one or more surfaces. The individual waves - as well as their sum - are highly dependent
upon the frequency of the propagating waves, their path lengths as well as the reflective
properties and geometric arrangement of the encountered surfaces.
This research will exploit the assumption that multipath fading of a signal from a
single source results from the sum of many reflected waves. The individual waves have
roughly equal power as well as independent amplitudes and independent phases. This
allows the fading component of the incident signal to be modeled as a complex Gaussian
random variable with uniformly distributed phase and an envelope which is Rayleigh
70
distributed.'"' If the channel is well-approximated by a constant frequency response
characteristic then relative time delays between arriving wave fronts are negligible. This is
known asfrequency-nonselective Rayleigh fading or flat Ray leigh fading. The flat-fading
condition is probably an accurate approximation for indoor communication systems with
large path losses and mobile systems with scatterers located in close proximity to the
mobile. It represents a worst-case condition from the standpoint of detection since it will
not allow the use of a RAKE receiver*^ to provide resolution of individual time-delayed
paths.
The complex base-band output of a sensor array outwardly resembles the expression
given in equation (4.1):
y(/) = AilO'--.c„(.-Tjfc„(r-T„K-Hn(r) (5.1)
where the elements of the phase shift vector Um are complex Gaussian random variables
which result from Rayleigh fading. This is in contrast to equations (1.4) and (4.1) in which
the components of Um are complex exponentials resulting from directional, unfaded
signals.
The components of Um may have any permissible degree of correlation. The study of
spatial diversity combiners is dedicated to antenna array structures which force the
correlation of fading components between array elements to be low (ideally zero).'"* A
base station diversity array must have larger element spacings than the customary half-
wavelength spacings used in a mobile radio receiver.^* Lee conducted an empirical study
of fading correlations in a two-element base station array. He concluded that, for low
correlation, the interelement spacings must be 15X, - 20A, if the signal arrives from
71
broadside and 70A. - 80X, if the signal arrives along the axial direction. Salz and Winters'**
examined a linear array and developed closed-form expressions for the direction-
dependent fading correlations between array elements when multipath rays are "dense"
throughout a range of DOAs. Raleigh et al."° proposed an analytical model which
describes the spatially-dependent correlations of the fading process. Verification of the
latter two models through experiment remains an open issue, as does more refined
spatially-dependent channel models. Naguib and Paulraj*^ examined the effects of the Salz
and Winters fading model on a base-station diversity array in an IS-95 system using
closed-loop power control.
Because spatial channel models remain an open issue this research will exploit the
assumption that the fading process is independent between antenna elements and the
elements of Um in equation (5.1) are complex i.i.d. N(0,1). Spatial dependence of the
incident signals - via the interelement phase-shifts - no longer exists and the array
geometry is critical only in that it results in independent fading between elements.
Wmters has shown that even when directional information is not used by the processor
(Le. the array functions as a diversity combiner) an optimum combiner will outperform, in
steady state, a maximal ratio combiner because it is able to adaptively attenuate cochannel
interference.
Multiple Cells with Signals Subjected to Rayleigh Fading and Shadow Fading
The models and results from the last subsection will be extended here to include the
effects of outer-cell interference. Like the previous section of this chapter the incident
signals will be subjected to flat Rayleigh fading. Unlike the previous section, however, the
72
cell which contains the desired user wUl be surrounded by several layers of cells containing
sources of multi-access interference (i.e. other DS CDMA users). The interference will be
subjected to multipath fading and shadow fading. Shadow fading occurs when structures
(such as buildings, hills or mountains) attenuate propagating signals. Shadow fading varies
more slowly than the multipath fading component of the signal and is interpreted as the
time-varying mean of the rapid, multipath-induced signal fluctuations.^*
The incident signal model may be modified sUghtly:
K
1
m=l
y{t) = AZIO'-'^" • c„(r - t„)b„{t - tJu„ +
f^nsj20\ (5.2)
V./20
V ''".<> J
where the first summation is for center-cell and the second summation (with index n)
results from the outer-cell interference. All users have lognormally distributed power
control error where em is N(0,OpJ). The quantity Noc is the number of outer cells while K
remains the number of users/cell. The variable Sn is N(0,64) which in mm specifies
lognormally distributed shadow fading with 8 dB standard deviation. The quantity rn.o* is
the distance-dependent, fourth-order propagation loss between the n'th interferer and the
center cell base station. The phase-shift vectors u^ and u„ are, as in the previous
subsection of this chapter, composed of i.i.d. random variables which are N(0,1). Note that
the incident amplitude A„ is an indexed quantity unlike the first summation representing
the center-cell users. This is because the outer-cell amplitudes are determined by a hand-
off to the outer-cell base station with the least path loss. This will be discussed in more
detail shorUy. The quantity n(t) is a vector of complex AWGN with power o^.
73
Figure 5. 1 Spatial region for simulation of outer-cell
interference.
Figure 5.6 below shows a diagram of the arrangement of the hexagonal cells, each
with a unit radius ( cell area = 3v^/2 ). The region of interest is circular and contains the
equivalent area of Noc = 21.67 cells, excluding the center cell. The small circle at the
center of each cell designates the position of each cell's base station. The wedge-shaped
region within die larger circle contains the equivalent area of 3.61 cells and it is over this
region that the spatial distribution of outer cell interferers is uniform, excluding the portion
containing the center cell. The interference from this wedge-shaped region will be
determined for each scenario and the results replicated to generate the interference
components for the remainder of the circular region.
The path loss which occurs between an outer-cell user and an outer-cell base station
contains a deterministic propagation loss (« r'' ) as well as lognormally distributed
74
shadowing. The random components complicate the hand-off - or membership - of a user
to a cell: a user is not necessarily serviced by the closest base station. Viterbi et al.'"
investigated the properties of outer cell interference for the case of perfect power control
and lognormal shadowing while Lee et al."^ investigated the effects of imperfect power
control and lognormal shadowing. These previous works examined the case of up to 4
base stations involved in the hand-off. Increasing the number of base stations in the hand-
off beyond iVs = 4 will only slighfly decrease the outer cell interference for a given degree
of shadowing and power control error.
The hand-off was computed by selecting the base station path with the minimum
base-station-to-mobile path loss. For the mth outer-cell user, this results minimizing a
convenient ratio of path losses between the outer cell mobile and the outer cell base
station and the outer cell mobile and the center-cell base station:
mm
'
»-..
lO'
10
*-,.
«„
A.
•10
10
•10"
To
where: m = L..KN„ (5.3)
where m represents the index over the outer-cell users in the wedge-shaped region and n is
the index over the //b =11 base stations denoted by the solid black circles in Figure 5.6
above. The quantities r^^ and 10'--"^'° are the propagation loss and lognormal shadowing
respectively between the mth mobile and the nth base station. The quantities r^^ ,
10"*-" " and are the propagation loss and shadowing between the mth mobile and the
center cell at the origin. The quantity 10"'-'"° is the power control error of the mth user
with respect to the base station chosen for hand-off. As the figure shows, base stations just
75
outside of the wedge-shaped region are utilized in the hand-off calculations for the outer
cell interferers. The shadowing and power control error components are assumed to be
independent from path-to-path. Some authors"^'""* maintain that spatial correlations exist
in shadowing components, but those effects are not incorporated into this model. The
incident levels (A„'s of equation (5.3)) resulting from hand-offs in the wedge-shaped
region of Figure 5.6 were replicated to generate interference levels for the remaining
portions of the large circular region. The model used in this research agrees precisely with
the results of Viterbi et al.'" which reported that for Nb = 3 the ratio of the outer cell to
inner cell interference is 0.57.
It was assumed that power control error correlates perfectly between hand-off base
stations and does not enter into interference calculations until hand-off is chosen based on
minimum path loss. This is in contrast to the work of Lee et al."^ which assumed closed-
loop power control error was uncorrelated between hand-off base stations. The authors
minimized the quantity:
mm
4,=
n = I...Nb
V.
«-..
m,n
•lO"
10
•lO"
10
V.
c
•lO"
10
where: m = L..KNg (5.4)
where 10 '° is the power control error between the mth user and the nth hand-off base
station. They assumed that Nb =3; hand-off occurred to one of the three closest base
stations.
76^
Analysis
This section will briefly revisit the analytical results of chapter four which gave an
asymptotic expression for capacity when signals were directional. Another analytical
model will then be introduced which exploits the assumption of lognormal interferers.
The expression given in equation (4.3) for the {Eh/No) eg may be rewritten slightly
using equation (4.4) for a single cell:
N.SNRifip.W^^' ^^^^
where 5A«/„ =(A^/ct^)-10'-^'° is the input SNR and Gp„ =|w>„| /||wj| is the
normalized array gain towards the mth signal. In chapter four some assumptions (via
equation (4.4)) allowed the sum term in the denominator of the expression for (Eb/No)^
to simplify into products of average terms, some of which are not explicit functions of the
input parameters (i.e. avg{Gpm } = Gpi, m ^t 1, the interference signal gain). In addition,
the gain towards the desired user Gpi = Gpd was modeled by its upper bound A^, the
number of array elements. While these approximations were not accurate when considered
separately, their quotient resulted in a convenient form and gave acceptable results ( see
Figures 4.2 and 4.3). The per-element capacity was the slope of equation (4.8):
„ 3NN,
' ln(10)Q-'(Pw) inilO , '
10 '^ 200 '^
(5.6)
where the term containing the Q-function is dominant for the range of power control error
considered here. The term - analagous to a single-pole roll-off - resulted from modeling
77
the multi-access interference as an averaged quantity. The averaging operation left the
desired signal as the only random variable and resulted in a lognormal distribution of
(Eb/No)eff in which the standard deviation (in dB) is determined solely by the power
control error standard deviation Opce (see equations (4.5), (4.6)). Under this simplified
analytical model, the number of users K and the number of array elements N affect the
mean of (Eb/No)eff but not the standard deviation.
This series of assumptions gave acceptable results for the asymptotic case of strong
nominal input levels if the signals were directional and had a moderate degree of power
control error. With the addition of multipath fading the results of chapter four are less
precise. The roll-off of a diversity combiner's per-element capacity is slower for low
values of power control error because the standard deviation of (Eb/No)eff is not
determined solely by the power control error Opce.
A different analysis technique which might approximate the standard deviation of
(Eb/No)eff more accuately has been examined by other authors. The model assumes that a
sum of lognormally distributed random variables is also lognormal. Along this line,
Schwartz and Yeh developed an iterative version of Wilkinson's method"^ and then used
their model to evaluate the outage probability of a multi-cell AMPS system."^ Beaulieu et
al. examined several methods for approximating a sum of lognormal random variables and
concluded that the "best" choice of model depends upon the system parameters (i.e. the
degree of shadowing and the magnitude of the outage probability).
This research will supplement the analysis of previous sections by approximating the
multi-access interference as a sum of lognormal r.v.'s. This method will result in a closed-
78
form expression for outage probability. An explicit equation for the per-element capacity
will not be possible.
If the array gain towards the interference may be modeled by a constant, the resulting
noise and multi-access interference is approximated by:
/j^ = yio'° +^^yio'°+ — 2 — (5,7)
±i Gp; ± IGp.SNRi
where the left-most summation represents the inner-ceU interference and the second
summation represents the outer-cell interference. The right-most term results from the
presence of noise. The outer-cell interference has been modelled as a quantity normaUzed
by the inner-cell interference: the outer-cell summation is over K rather than KNoc. This
approach has been used for incident signals by previous authors"'"^ and will be extended
to the diversity array output via the gain constant F.
Wilkinson's method begins with the assumption that the multi-access interference is a
lognormal r.v. The first two moments of the Ima are then matched to the first two
moments of the sums of lognormal random variables. The mean and second moment oil ma
are given in equation (5.8) below:
e[u] = e
expj^hidO)—
= exp{Pm,+P'(jl/2)
F / , \ 3
=(*:-.)exp(/J'.J/2).-^exp(/J'.,V2).^^^^ = *.
79
eIiL] = e
expl21n(10)—
= exp(2pni2 + 2p^ol)
= E
;iexpK) + ^iexpfe)^^^|^^
EllL] =
(K - l)exp(2P'G^) + iK- m - 2)txp{pW, )
r p\^
—-\(Kcxp(2p'ai)+KiK-l)cxp(pW^))
Gp,
+ 2
Gp,
GpiSNRi
/i:(/^-l)exp(/3^(c7,^+(j,^)/2) + |
{K-l)exp{P'G'j2)
^2
2GpSNRi
3F
GpfSNRi
Kcxp(p'al/2)
= b, (5.8)
where the variable z is N(m, , a/> and )9 = ln(10)/10. Note that the exponential terms of
the inner and outer cell interference are specified as N(0,cf^) and N(0,Gy^) respectively.
The logarithms of the moments allow linear solutions of m^ and (5^ in terms of the
moments of the sums {bubz):
'«z=21n(Z.,)-iln(feJ
CT^' =ln(Z;2)-21n(Z?i)
Once the interference moments are determined then (Eb/No)effmay be expressed as:
(5.9)
80
(e ^
y^oJeff
x,-101og(exp(l))c
= ^N.
10
10
(5.10)
and the outage probability is given by equation (5.1 1) below where log(») denotes the
base- 10 logarithm. This method does not allow an expUcit closed-form expression for the
per-element capacity. It will, however, allow that quantity to be extracted from outage
probability contours as was done for the simulated results. The expression for the outage
probability from the approximated moments:
Pr
r T^ \
. N
\\"<>-'eff.df
where:
77 = 10*log
<^B
= Q
r/-4e-101og(exp(l)H
^3„Gp/
V^
Gp,
= 10* log
2 'Gp,
G,=^al + \m\og\tx^(l))al
(5.11)
Results
Most of the details of the simulation procedure are outlined in the section Simulations
and Results of Chapter 4 with one difference: the faded, incident signals were no longer
directional and a DOA was not specified. Each signal was specified by its power control
error (and shadowing for the outer cell interferers), and its phase-shift vector which
consisted of complex, i.i.d. unit-variance Gaussian random variables. For each trial the
(Eb/No)eff was determined via equations (1.8), (1.10),(5.3) and (5.5) and was then
compared to an outage threshold of 7 dB. A sample mean was tabulated over the trials
for each combination of users and array elements. Even with fading the resulting outage
81
contours were roughly linear in N with slopes which decrease with increasing power
control error.
Figure 5.2 shows Pr((Eb/No)eff < 7 dB) versus the number of users with the power
control error equal to 4 dB and with no outer cell interference. As in previous plots, the
curves are plotted parametrically. The different curves represent outage probability for
different numbers array elements N. For this value of power control error the performance
is not drastically different than for the nonfaded case shown in Figure 4.4. The nominal
input signal levels (without power control error) are such that Eb/No = 7 ( or SNRi = 7/Nc
= 7/127 = -12.6 dB) for a single signal incident on a single array element feeding a
Figure 5.2 Outage probability versus the number of incident signals with power control
equal to 4 dB. The number of array elements A^ is a parameter.
82
conventional detector. When comparing Figures 4.4 and 5.2 it is easy to see the faded and
unfaded systems have roughly the same outage performance. This similarity ceases for
other values of power control error.
Figure 5.3 shows simulated outage contours as the power control error varied. The
top two plots show contours of Pr((Eb/No)eff < 7 dB)= 0.02 as the power control error
varied from to 7 dB with no outer cell interference. The upper plot (a) is for a nominal
input condition (without power control error) of SNRi = 7/Nc = -12.6 dB while plot (b)
shows contours for a nominal input condition of SNRi = lO^/Nc = 59 dB. Plots (c) and (d)
show the same conditions but include outer cell interference. The plots show that, even
with fading, outage-based capacity continues to be linear in ^V. Note also that as the
nominal input levels increase, the intercepts of the approximately linear contours move
towards the origin. This general behavior was predicted for the nonfaded case by equation
(4.8).
The slopes of the lines in Figure 5.3 were extracted using a least-squares curve fit and
plotted as the per-element capacity in Figure 5.4. Note that tiie per-element capacity is less
than that for the case of no Rayleigh fading as long as the power control error is less tiian
approximately 1.5 dB. As long as tiie power control error is greater than 1.5 dB the
receiver employing a diversity array combiner in the presence of fading outperforms a
receiver using a beamformer to equaUze signals which are not subjected to multipaUi
fading.
83
15
(a)
T-
4
(b)
20 25 30
Number of Array Elements
5 6 7
Number of Array Elements
5 6 7
Number of Array Elements
Figure 5.3 Capacity versus the number of array elements with power control error
as a parameter. Outage probability = 0.02. Plots (a), (b) are without outer cell
interference. Plots (c), (d) include outer cell interference.
84
Note that in Figure 5.4 there is degradation in the per-element capacity as the nominal
input levels increase and outer cell interference is present (plot (b)). When the nominal
signal level is small (SNRi = -12.6 dB) the individual outer cell signals incident on the
inner cell are very weak (compared to ambient noise and the inner cell interferers) due to
the path loss. In this circumstance the outer cell interference probably just adds to the
ambient AWGN. When the nominal input signals are large {SNRi = 59 dB), outer cell
interferers can overcome the outer-cell to center-cell path loss and can have signal levels
much higher than the ambient noise, even at the center cell. In this case some percentage
of individual outer cell signals compete with the inner cell interference for attention from
the array processor, and performance suffers.
Per-
Element''
Capacity
Per-
15
Element
Capacity
10
1 I
e O — G SNRi = -12.6 dB
SNRi = 59.6 dB
1
I
1 r-
i ^
• .•
-G-
-9 SNRi =
= -12.6dB
= 59.6dB
"""""-"!•"-
~~-S^
•
— SNRi =
i 1
^^^^^f-^-i^-^^^
3 4 5 6 7
0^) Power Control Error (dB)
Figure 5.4 Per-element capacity (users/array element) versus power control error
(dB) for an outage probability = 0.02. The upper plot is for the case of no outer
cell interference. The lower plot includes outer cell interference.
85
Figure 5.5 below shows the per-element capacity from the simulated results of Figure
5.4b (with outer cell interference) and the approximation given in equation (5.2) with ^ =
7 dB and Gpi = 2. Agreement between (5.2) and the simulated results for the case no
outer cell interference was poor, and the results are not presented here. The value of Gpt
accounts for the fact that the output power of the outer cell interference is about equal to
the output power of the inner cell interference when averaged over most of the possible
combinations of user population, array elements and power control error. The rapid roll-
off of the curve from equation (5.2) is due to the use of averaged multi-access interference
when computing (Eb/No)eff, as discussed in a previous section.
Figure 5.6 shows the per-element capacity resulting from the application of
Wilkinson's method to model the multi-access interference as a sum of lognormal
variables (equations (5.7) through 5.(1 1)). The lognormal statistics of the inner cell
interferers are assumed to be due to power control error only while the outer cell
interference combines power control error and 8 dB of shadowing {xm of equation (5.7) is
N(0, Opce^)) while y„ is N(0, q^ + 64)). Simulations show that the average interference
gain Gpi for the inner cell interferers can vary from zero to imity depending on the
number of elements, the number of users and the nominal input level. Interestingly enough,
Gpi does not seem sensitive to the power control error. Averaging over these conditions
results in Gpi » 0.7. The gain constant for the outer cell interferers - F of equation (5.7) -
is taken to be unity. Simulations have shown that the array attenuates the interference so
that the average output power due to the inner and outer cell interferers is about equal
86
£.\)
'
'
1 1
1 1
18
-'•.
•f ♦••
• Eqn. (5.2)
16
Per
•
*
'•.
-© 9-
Simulations,
SNRi= -12.6dB
Element14
C^)acity C
;^
•
V.
•
Simulations,
SNRi= 59 dB
12
■
■
10
8
6
"^
"^V
•
4
•
^^^^^S^
•
2
•
^*'*'^^**t>^^
n
1
^^^"^^^^^^^n
1
2
3 4
5 6 7
Power Control Error (dB)
Figure 5.5 Per-element capacity versus power control error (dB) with outer cell
interference. The capacity is from equation (5.2).
over many scenarios of interest. Interestingly enough, simulations show the ratio of the
incident signal power from the outer cell and inner cell sources is 0.57.
The upper two plots of Figure 5.6 show that Wilkinson's method does not provide a
particularly useful approximation when there is no outer cell interference. The lower two
plots include the effects of outer cell interference and tend to agree more closely with
simulated results for a range of power control error from 1 to 6 dB. For power control
error = 7 dB the simulated per-element capacity is almost an order of magnitude greater
than the analytical results.
87
Per 30
Element
Capacity
20
10-
1
Per 30
Element
C^jacity
20
10
■I 1 1" 1
^>.^. .:. ; ;
I 1
SNRi = -12.6 dB
Analysis
^~^J i i
i i i
4
(a)
1
i ,
1
-:
^...'^i.^.
■]■■■■
I
SNRi = 59 dB
" Analysis
i
t~
•^^^TtQ:^
J '
^ ^
4
(b)
Per 15
Element
Capacity
10
Per
Element
Capacity
5 6 7
Power Control Error (dB)
Figure 5.6 Per-element capacity versus power control error (dB) for outage
probability = 0.02. Plots (a) and (b) are without outer cell interference. Plots (c)
and (d) include outer cell interference. Curves from analysis result from equations
(5.8) -(5.11).
88
Conclusions/Discussion
Simulations showed that even in the presence of multipath fading, the per-element
outage-based capacity might serve as a useful performance measure for a wide range of
conditions when considering a single-cell with multi-access interferers. As the nominal
input levels increase for the multiple-cell case the per-element capacity deteriorates for pee
^ 3 dB compared to the single-cell case. The reason: for low nominal input levels the array
treats the outer cell signals like AWGN. When the nominal input levels are high all
interferers incident on the base station are well above the noise and the array must
r dedicate some processing to attenuate the outer cell interferers, at a loss in performance.
The simulated results presented in this chapter show that for low nominal input levels
(SNRi = 7/127 = -12.90 dB) and power control error > 1.5 dB a single-cell system with
multipath faded signals will have a higher per-element capacity than a system with
directional, unfaded signals. The addition of outer-cell cochannel interference
degrades the per-element capacity by about 80-90% for p.c.e. < 2 dB and 60-70% for
p.c.e. > 2 dB. For strong nominal input levels {SNRi = 59.6 dB) the addition of outer cell
interference degrades the per-element capacity by a factor of 2/3 to 3/4 for power control
error < 2.5 dB.
The two analytical models for per-element capacity gave mixed results. The model
developed in Chapter 4 - which averaged multi-access interference - degrades too rapidly
with increasing power conttol error to be of much use in a single-cell scenario. In a multi-
cell scenario witii low input levels the model gives a good fit to simulated results for
power control error > 1 dB. When input levels are high, tiie array output power due to tiie
89
outer cell interferers can be as much as twice the power of the inner cell interferers (there
are 21.67 times more signals originating in the outer cells than the inner cells). This
accounts for the decrease in the capacity in Figure 5.5 as the nominal input levels increase.
This model is limited in several ways. First, we are modeling the interference as an
averaged quantity so the complex trial-to-trial interactions between the interference, the
desired signal and the noise are lost. Second, we are attempting to absorb the relatively
complicated behavior of the processor into two parameters, Gpi.and N, which do not vary
with the number of users or the power control scenario.
The model based on Wilkinson's method of evaluating sums of lognormal random
variables was introduced in an attempt to resolve the first of these two issues. It was
hoped that the accuracy of the analytical model might be improved by approximating the
interference as a sum of variables rather than an average. The plots in Figure 5.6 show that
this model might offer some improvement in accuracy - compared to simulations - for the
select case of multiple cells. If more accuracy is required for the case of a single-cell
system or power control > 6 dB, then refinements of the models would be necessary.
Summary
This chapter presented simulation results of the outage-based capacity for incident
signals subjected to frequency-nonselective Rayleigh multipath fading and lognormally
distributed power control error. The models included single-cell and multi-cell scenarios
where outer cell interferers were subjected to lognormal shadow fading. The multipath
fading model assumed that the spatial interactions between the faded incident signals and
the antenna array allowed the fading process to be fully decorrelated between array
90
elements. Under these conditions the array functioned as a diversity combiner rather than a
beamforming antenna array.
The per-element capacity - the number of users/array element which may be
supported for a required outage probability - was evaluated via simulations. The
motivation behind the use of per-element capacity was to find a performance measure
which was robust to the variations in user population and array size and which would
reflect the possible contributions a receiving array would make to the capacity of a power-
controlled CDMA system.. An additional goal was to formulate approximations which
would allow easy assessments of the per-element capacity.
Physical arguments in conjunction with semi-empirical curve fits resulted in two
analytical models. The first model was adapted from the model in Chapter 4 in which the
signal, the interference and the noise were averaged. The complex interactions between
even these quantities was overlooked for the sake of simpUcity. Array outputs were
modeled as averages of input quantities and gain constants. Values of these constants were
derived empirically from simulations.
The second model exploited Wilkinson's method of approximating lognormal
variables so that averaging of the interference could be avoided. This model resulted in
sUghtly improved accuracy - compared to the first model - for power control error < 2 dB
but was somewhat worse for power control error > 6 dB.
CHAPTER 6
BASE STATION ANTENNA ARRAY ADAPTIVE PERFORMANCE
This chapter examines the steady-state performance of the Recursive Least Squares
(RLS) adaptive algorithms for a base station diversity array receiving faded incident
signals with power control error. A simulation approach is used since the scenarios are too
complicated to allow useful analytical solutions. The simulations were discrete-event
simulations of transmitted bits through a nonstationary AWGN channel with time-varying
multipath fading as well as stationary power control error and shadow fading. The
simulation results will be given in terms of histograms of the effective Eb/No which can
facilitate outage calculations. The performance of an RLS array in the presence of
cochannel interference has been examined by several authors,^^-^^" but of these only
Tsoulos et al. considered near/far scenarios with fading and outer cell interference. They
used simulations to determine the outage probability when the (A^ = 8) array functioned as
a beamformer, not a diversity combiner.
Simulation results show that the RLS algorithm can track the time-varying solution
effectively and that the adaptive solution is close to the steady-state solution. The
structure of this chapter is similar to the two previous chapters. The first section revisits
the now-familiar signal model from previous chapters and also introduces the time-
dependent fading process. Approximations of the fading process used in this research are
also presented. The second section outlines the RLS adaptive algorithms and the receiver
91
92
structure used in the simulations. The third section gives the results of the discrete-event
simulations and provides some discussion. The last section gives the conclusions.
Si gnal and Channel Model
The signal model is the same as in the previous chapter:
i;W = AilO'"-.c„(r-rJfc„(r-Tju„-h
IA„.10'"-. 10_ .c,(r-T,K(r-r„K+n(r)
where the first summation is for center-cell and the second summation (with index n)
results from the outer-cell interference when it is taken into account. Power control error
and shadow fading are lognormally distributed {e^ '\&N(0,apJ),Sx is N(0,64)). The
quantity Noc is the number of outer cells while K remains the number of users/cell. The
quantity r„ is the distance-dependent, fourth-order propagation loss between the nth
interferer and the center cell base station. As in Chapter five the amplitude A„ accounts
for the hand-off with the least path loss when outer cell interference is taken into account.
The quantities c^^ )an<ibx() represent the spreading code and the BPSK modulation with
ideal, square pulses. We assume the beginning and end of a FN sequence corresponds to
the beginning and end of an information bit. In this chapter the FN sequences, no longer
random, are Gold codes of length 127. The quantity n(t) is a vector of complex AWGN
with power a„^ which is temporally and spatially white. The power control error and
shadow fading are assumed to be constant over the observed interval.
93
Time-varying, flat Rayleigh fading, is introduced into the incident signals by the
phase-shift vectors Un, and Un. The fading process is independent between array elements
(spatial samples) but is correlated between time samples. The time autocorrelation
function is given by:
R{T) = JoicOoT) (6.2)
where ttb is the Doppler frequency in radians/sec, x is a time delay and Jo(*) is a Bessel
fimction of the first kind with order zero. The Doppler frequency arises from relative
motion between the signal source and the observation point; it represents a velocity-
induced shift in the signal carrier frequency. The Doppler frequency is given by ttb = ^
where k is the free-space wave number (k = liWk ) and v is the relative velocity.
The fading of the incident signals was accomplished by superimposing a time- varying
complex Gaussian component on the time-dependent data waveforms. The fading
component was generated by a third-order autoregressive (AR) filter. The filter response
to white noise has an autocorrelation function with a damped sinusoid characteristic which
has been adjusted to closely match the major lobe and first minor lobe of the Bessel
function in equation (6.2). The impulse response of the filter is given by:*"
M0 = (^^''^*''[2.66/oSin(5.15/o0-3.17/oCOs(5.15/or)-f-3.17e-"'^'''])M(r) (6.3)
where /d is the Doppler frequency in hertz and u(t) is a step function at r = 0. The filter is
used in the following manner. First, complex white Gaussian noise is processed with the
filter specified in equation (6.3). The data waveform is then multiplied point-by-point by
94
the filter ou^ut to simulate the fading process. This approach preserves the salient
properties of the fading and allows an efficient simulation procedure.
Adaptive Receivers
The RLS processor has an advantage over the LMS algorithm in that its convergence
rate is not dependent upon the eigenvalue spreads of R, the input autocorrelation matrix.
The disadvantage of this algorithm is that the computational burden (0(N^)) is relatively
large compared to the computationally efficient Least-Mean-Squares (LMS) algorithm.
Unfortunately an LMS array cannot track the time-dependent fading, especially at higher
vehicle speeds. "^^
processor 1
Figure 6.1 Adaptive array with a discrete-time conventional detector.
95
The discrete-time adaptive receiver structure for user 1 is shown in Figure 6.1 below.
The outputs of the antenna elements are passed through integrate-and-dump filters
matched to the chip transitions of the desired signal waveform. The sensor array outputs
are then weighted and summed. The array output is passed through a discrete-time version
of a conventional detector which consists of a delay-Une filter with Nc discrete time-delay
elements each with a duration of one chip (dotted box). Once every bit the contents of the
delay line are weighted by the elements of the PN sequence (Ci = [ci Ci Ci ...Ci " ] e
{ 1,-1 }) of the desired signal and summed to form the output The estimate of the current
output bit is taken as bi,est(n) = sign(Real{So,i(n)}). The array weights are updated once a
bit For the nth bit interval the input of the array and the weights are given by:
w(n) = [wo{n) w,(/j) • • • w^_,(/i)f
X(«) =
x°{n) xl{n)
(6.4)
^:-
■'(")■
^r-
■\n)
■*Af-l
"(«).
where Xg'"(n) represents the array input at tiie gUi element during tiie mth chip interval of
the nth bit. The inputs are processed by tiie integrate-and-dump filters. The input to tiie
bank of weights is given by:
Y{n) =
nT
hT-Tc
\xl{n)dt \xl{n)dt
nT-2Tc
hT-T,
hT
nT-(N,-\)Tc
x:^-\n)dt
hT-NcTc
nT-Tc
jxl_,{n)dt jxl,_,{n)dt
nT-7Tc
'x',i;'{n)dt
nT-T,
hT-N^T^
(6.5)
The inputs are weighted and summed by tiie array and tiie resulting output stream is
96
weighted and summed by the tapped delay-line. The array/detector attempts to minimize
the output error:
ein) = \Un) - w'(n)Y(n)c, = ft,„,(n) - w^(n)z(/j)
(6.6)
where the random variable is z(n) = y(n)ci. A single element of z(n) is the inner product
between the output of one array element - filtered chip-by-chip over the bit interval - and
the desired user's PN sequence Ci. The vector z(n) will be the sample vector over which
the adaptive filter operates.
The RLS algorithm recursively computes an estimate of the optimum weight vector
Wo. The recursive equations are given by (version 2, Haykin.^^):
^ y-'R-'(n-l)z(;z)
^ ^ l + r'z"{n)R-\n-l)z{n)
a{n) = r{n)-w"{n-l)z{n)
(6.7)
w(n) = w(/i - 1) -I- k{n)a'{n)
R-i(n) = y-'R-'{n - 1) - y-'k{n)z" {n)R-\n - 1)
where a(n) is the a priori estimation error and is an estimate of the error between the
current value of the reference signal r(n) and the output based on the previous value of the
weight vector at iteration n-1. The weight vector is updated by the product of k(n) and the
a priori estimation error. The vector k(n) may therefor be interpreted as a gain vector and
is sometimes called the Kalman gain. Note that the matrix inverse is updated recursively,
rather than by a block computation.
97
The quantity y serves to exponentially window the data. Eleftheriou and Falconer"*
investigated the influence of y in the adaptation process. They examined the role of y on
the total misadjustment, the sum of the "estimation noise," which results in the
displacement of the filter weight vector from the optimum value, and the "lag error" which
occurs when the filter tries to track channel nonstationarities. Too large a value of y will
cause a filter to use outdated statistics when tracking channel nonstationarities. An
excessively small value of y will allow fast adaptation but will cause too little information
to be used in the adaptation process and result in excessive misadjustment In this work
the selection of y was empirical. More details will be given in the next section.
In the adaptive receiver shown in Figure 6.1 the bit estimate besi(n) serves as the
reference signal r(n) for the RLS processor. The structure shown is for decision-directed
operation only. Training and transient response issues as well as issues arising from fixed-
point arithmetic were not addressed by this research.
Simulations
Monte Carlo simulations were conducted for the discrete-event case. Rather than
examining exhaustive combinations of power control error, user populations and array
sizes, discrete-event simulations focused on a few combinations of these quantities in an
effort to show that the adaptive solution does not differ appreciably from the steady-state
results presented in an earUer chapter.
Results in this chapter are presented in terms of the histograms of the (Eb/No)eff.dB
When (Eb/No)eff is tabulated in dB units for many trials the resulting histogram is
98
approximately Gaussian but contains some slight asymmetries. This result is not surprising
considering that the mean incident signal amplitudes are lognormally distributed."'"*^
Some authors have found that a sum of lognormally distributed random variables is well-
approximated by another lognormally distributed random variable.
All signal sources were assumed to have velocities of 55 mph relative to the base
station. This corresponds to a Doppler frequency of 74 Hz. A real mobile communication
system must cope with multiple signals Doppler-shifted over a range of values: zero for a
car sitting still or 1 14 Hz for a driver going 85 mph. Note that the Doppler frequency or
frequencies could have an appreciable impact on the selection of Yr, the RLS memory
factor.
Incident signal levels were modeled via the signal description given earlier in the
chapter (equation (6.1)). All incident signals consisted of random, equally-likely BPSK bits
(rate = 9600 bits/sec) which were spread with Gold codes of length 127 and chip-filtered
via the integrate-and-dump filters of Figure 6.1. The effects of multipath fading were
introduced via Um and Un - the independent array samples - in equation (6. 1) and the
multipath fading simulator described in an earUer section. The fading was generated at the
chip rate and was not considered constant over a bit interval. A single scenario consisted
of multiple signal vectors of 51 bits in which power control error and shadow fading were
constants and only the multipath fading component of the incident signals varied with
time.
The RLS processor was modeled via equation (6.4) above. In order to avoid
transient-related issues the initial weight vector for the RLS algorithm was the steady-state
99
MMSE weight vector averaged over the first bit Output signals were then tabulated over
the next 50-bit interval and (Eb/No)eff was calculated for each single bit. Six hundred
scenarios for a total of 30,000 RLS-adapted bits were simulated. The nominal input SNRi
= -12.6dB.
The choice of y was made empirically. A scenario was considered with 5 inner-cell
50 100 150 200 250 300 350 400
Time in Bits
Figure 6.2 The time-varying (Eb/No)eff for an RLS processor and an ideal MMSE
processor. There are 5 users with pee = dB incident on a 3-eleraent array. The
Doppler frequency is 73 Hz. and y is 0.825
users with pee = dB subjected to multipath fading. The (Eb/No)eff was tabulated on a
bit-by-bit basis for an interval of 400 bits using the output of an RLS processor
((Eb/No)eff,RLs) and the output of an ideal MMSE i(Eb/No)eff.MMSE) processor. For these
conditions a value of Xr = 0.82 was found to minimize the quantity:
100
I- J 400 „^i W No J ^,RLS \N0 Jeff MUSE J
(6.5)
and was used to produce the results that follow. Figure 6.2 shows the time-varying
(Eb/No)eff for the ideal MMSE processor and the RLS processor over the 400-bit interval.
The results of simulations for 5 users with Opce = dB is shown in Figure 6.3 below.
0.14
0.12
0.1
0.08
0.06
0.04
0.02 h
-30
-20 -10
Effective Eb/No in dB
N = 3
K = 5
pce = OdB
fD = 73Hz
- Y= 0.825
: :in= 10.98 dB
i i o = 3.46 dB
t
\^ ...
1 ,
: I
i i
I 1
\ —
i Iv
— —
1 i nfll
mm L
10
20
Figure 6.3 Bar chart of histogram of (Eb/No)eff in dB. The soUd line curve oudines
a Gaussian pdf with mean = 10.98, st. dev. = 3.46. There are 5 users with pee =
dB incident on a 3-element array. The Doppler frequency is 73 Hz. and y is 0.825.
The plot shows a bar-chart histogram of (Eb/No)eff in dB. Superimposed on this is
Gaussian curve with mean and standard deviation taken as sample means (m = 10.98, a =
3.46) from the data which generated the histogram. The match between the curves is good
except for the peak and the tail regions. Unfortunately, the lower tail region is critical for
101
outage calculations and the poor fit might prevent accurate results if a lognormally
distributed output is assumed and the outage probability is low.
Another interesting feature of Figure 6.3 is the slight asymmetry between the upper
tail regions of the bar-chart curve. The asymmetry is also present in histograms generated
by steady-state simulations. Using the histograms to calculate the outage probability gives
Pr{(Eb/No)eff ^ 7 dB) = 0.105, .125 for the bar-chart and Gaussian curve respectively.
Under the same conditions an ideal MMSE processor gave a mean output of (Eb/No)eff =
12 dB, a standard deviation of 3.27 dB and Pr(,(Eb/No)eff ^ 7 dB) = 0.066. Figure 6.4
shows the same conditions except that power control error has been increased to 4 dB.
Note that the mean is only slightly decreased to 10.7 but the standard deviation is
increased to 5.37.
Histograms which include the effects of outer cell interference are shown below.
Figure 6.5 shows the histogram of (Eb/No)eff for the case of perfect power control in
which outer cell users are subjected to 8 dB of lognormal shadow fading. The assignment
of the power levels for the outer cell users is the same as described in Chapter 5. As in the
previous plots a Gaussian distribution with the same mean and variance as the bar chart
histogram is shown. The mean ( = 10.6 dB) decreased sUghtly compared to the case of no
outer cell interference (= 10.98 dB ). The standard deviation (= 3.33 dB) decreased
slightly from a value of 3.46 dB for no outer cell interference. Using the empirical
histogram results in an outage probability of Pv{(Eb/No)eff ^ 7 dB) = 0. 15, a sUght
increase compared to the case of no outer cell interference. This seems to agree with the
general trends noted in the steady-state results which show only a slight degradation in
102
capacity when outer cell interference is included. The case of pee = 4 dB is shown in
Figure 6.6. The mean of the empirical bar-chart histogram is 9.75 dB and the standard
deviation is 5.17 dB. Unlike the previous figures, these curves resulted from 10000 bits of
data.
Conclusions/Summary
This chapter has examined via simulation the outage performance of an adaptive
diversity antenna array updated with the RLS algorithm. The output performance was
tabulated by histograms of (Eb/No)eff in dB. The adaptive performance did not vary
drastically from the steady-state performance. It appears that the RLS algorithm operating
in a decision-directed mode is capable of tracking the time-varying channel when Rayleigh
multipath fading is present
103
Figure 6.4 Bar chart of histogram of (Eb/No)eff in dB. The solid line curve oudines
a Gaussian pdf with mean = 10.98, st. dev. = 3.46. There are 5 users with pee = 4
dB incident on a 3-element array. The Doppler frequency is 73 Hz. and y is 0.825.
104
-5 5
Effective EbNo in dB
15 20
Figure 6.5 Bar chart of histogram of (Eb/No)eff in dB for the case of outer cell
interference. The solid line curve outlines a Gaussian pdf with mean = 10.6, st. dev.
= 3.33. There are 5 users with pee = dB incident on a 3-element array. The
Doppler frequency is 73 Hz. and y is 0.825.
lOS
0.12
0.1
0.08
0.06
0.04
0.02 -
N = 3
K = 5 users/cell
■ pee = 4 dB
fD = 73Hz.
Y= 0.825
1 -t 1 —
• ■ ■
« • I
• t •
L 1 n..
m = 9.75 dB J
: a = 5.17dB -\
I . . J II
.........
Vj
[
i >>
7,
i
t
»
yuSffifOulllll
1
-30 -20 -10 10
Effective Eh/No in dB
20
30
Figure 6.6 Bar chart of histogram of (Eb/No)eff in dB for the case of outer cell
interference. The solid line curve outlines a Gaussian pdf with mean = 9.75, st.
dev. = 5.17. There are 5 users with pee = 4 dB incident on a 3-element array. The
Doppler frequency is 73 Hz. and y is 0.825.
CHAPTER?
SUMMARY
In this work we have examined the possible performance contributions that adaptive
antenna arrays can make to DS-CDMA base station performance. This final chapter will
provide a results-oriented summary of the previous chapters. In addition, a section entitled
Areas for Future Work will outline some of the issues which have arisen during this
research which are related to array processing and channel modeling.
The system model has consisted of an array of antennas whose weights are controlled
by a MMSE processor. This work has examined two system configurations. First, a multi-
user antenna array was investigated which consisted of an array of sensors and a single
beamforming processor. The processor with a single set of weights attempted to equalize
the multiple incident signals. The second structure used an array of sensors which was
shared by multiple processors, each dedicated to a single incident signal. Each distinct
output fed a DS conventional detector.
Chapter two examined the steady-state performance of the multi-user processor.
Unlike most analyses of MMSE filters, this analysis focused on output SNR as the
performance measure. First, the case of K <N-1 spatially orthogonal users was examined,
followed by the case of 2 incident signals with an unspecified DOA spacing. The analysis
showed that for strong incident signals the output SNRs of the various signals were
leveled to a value near that of the weakest user.
106
107
Chapter three investigated the steady-state adaptive performance of an LMS multi-
user processor and attempted to establish a link between the misadjustment of the LMS
algorithm and the output SNR performance. The performance was determined by
examining first two moments of the output SNR. The first performance measure was a
cost function based of the difference between the mean SNR response of the processor
and the steady-state output performance. An upper bound for the cost function was
formulated which limited the excursion of the mean output SNR from the ideal steady-
state value. The second cost function was the quotient of the output variance and the
mean squared. Analysis and simulations showed that both cost functions were found to be
held to acceptable limits if the step size was selected using the well-known rules based on
convergence argimients of the LMS algorithm.
Chapter four examined the steady-state performance of a single-user processor. The
performance for this and the remaining chapters was judged on the ability of the array to
recover the desired signal and reject multi-access interference. This chapter attempted to
provide some insight into the contributions an array could make to the base station
receiver performance when multi-access users limited their transmit power via imperfect
power control. Rather than performing a rigorous analysis, an approximate analysis -
which replaced the interference response with its average - resulted in simple expressions
for the outage-based capacity. The outage-based capacity was found to be Unear in a
quantity which was the product of the processing gain and the number of array elements.
A robust performance measure - per-element capacity - was defined as the slope of this
line with the processing gain held constant Analytical results were corroborated with
exhaustive simulations.
108
The next chapter introduced multipath fading and outer cell interference into the
steady-state channel model. The approximate analysis based on the techniques of chapter
four gave mixed results. The approximate analytical model based on techniques from
Chapter four did not provide a good fit to simulated results for the per-element capacity in
the case of a single cell. The same model agreed well with simulated results for the
multiple-cell case as long as the power control error exceeded 2 dB and the nominal input
level was low (SNRi = -12 dB). Another model was introduced which approximated the
array gain by an average. The incident signals themselves were modeled using Wilkinson's
method of approximating sums of lognormal variables. The resulting expressions for the
per-element capacity were more accurate than the models from chapter four, for the
faded-signal case. When the asymptotic case of strong incident signals was considered, the
analytical expressions for per-element capacity gave slightly optimistic results compared to
simulations for power control error less than 4 dB.
Chapter six examined the adaptive performance of the MMSE processor via
simulation using the RLS algorithm. Performance was expressed through histograms of
(Eb/No)eff,dB and Gaussian curves with same mean and variance as the simulated data. It
was found that the mean and variance of the adaptive results agreed closely with those of
the steady-state simulated results. Unfortunately, the histograms were slightly skewed,
with the peak higher than the sample mean and a lower tail more pronounced than the
upper tail. The ou^jut was not exactly lognormal and that the skew might prevent accurate
analytical assessments of outage performance if the outage probability is small.
109
Areas for Future Work
Two issues have arisen during the course of this research which might warrant further
investigation. The first issue is the application and extension of previous woric to provide
rigorous, useful measures of array performance, particularly when then number of users is
large. The second is the extension of existing channel models to include the effects
spatially-dependent multipath fading.
This research provided semi-empirical approximations as convenient - but limited -
measures of outage-based performance. A more rigorous analysis is warranted. The
analysis might best focus on the joint statistics of the desired signal, the multi-access
interference and the noise. One possible approach is to find the pdf of the SINR:
SINR = s^'Q-'s* = tr(sVQ-' ) (7.1)
where Q is the covariance matrix and s is the signal vector of the desired signal. K signals
are subjected to independent Rayleigh fading then phase-shift vectors will consist of zero
mean jointly Gaussian i.i.d. elements. The joint distribution of the multi-access interference
components of Q are known to be Wishart distributed."' It might be possible to
formulate a pdf using these statistical relationships. This problem was noted, but not
solved, by Dlugos and Scholtz.'^"
Using previous work resulting from investigations into the Sample Matrix Inverse
processor (SMI) might provide a good starting point. The SMI processor uses a sample-
mean estimate of the covariance matrix Q:
1 '^
Q = t;Xx„x; (7.1)
no
to compute an estimate of the weights, w^ = Q"'s* , where x^ is the array output vector
of jointly distributed complex i.i.d. N(0,ax) variables at the mth time sample. The vector s
is the A^-length, phase-shift vector of the desired signal and consists of unit-amplitude,
complex exponentials. Obviously M > N so that Q is full rank. Previous woric has
examined the relationship between the window length M and functions and various
functions of the output power for stationary'^''" and nonstationary noise.'" It has been
shown that the output SINR normalized by its steady-state value is a random variable with
a beta distribution'^' for the case of stationary noise.
Reinterpreting this previous work may lead to some convenient results for diversity
combining. Recall that the samples that led to Q were interpreted as samples of the
Gaussian, zero-mean interference and noise. If the samples are instead phase-shift vectors
of Rayleigh-faded multi-access interferers, then Q may still be interpreted as a sum of
outer products of jointly Gaussian vectors. The results of some previous radar-oriented
research might be reinterpreted in terms of diversity combining. If a meaningful,
unambiguous measure of the output SINR of an optimum combining array is possible,
based on previous results, it might provide a robust solution to the problem of calculating
array performance when the number of users exceeds the number of array elements. This
is currently an active area of research. '^'*'^'
The second area of future work is the extension of general channel models to include
the effects of spatially-dependent multipath fading. A great deal of work has been done in
the area of temporal single-channel fading models but relatively few models exist which
include spatial effects for multi-channel receivers. The interest in antenna arrays as a
Ill
component in commercial communication systems has grown quickly in recent years.
Unfortunately, developments in array theory have outpaced the channel models. For
antenna arrays to reach their full potential as a communication system component
accurate, useful models must be developed which provide a unified, integrated description
of the temporal and spatial characteristics of the channel.
One such model is an extension of the single-channel with multipath fading which
assumes that multiple plane waves originate from the same source but arrive at a point in
space via different paths. In the single-channel model the effects of the individual waves
are modeled by "rays" which result in a Rayleigh-distributed amplitude and can result in
relative time delays between same-source arrivals. This model has been crudely extended
to include spatial effects by assuming individually faded, dominant rays arrive at points in
space from different directions. This type of model focuses on the distribution of large,
dominant reflectors in the environment about the receiver but ignores the smaller-scale
spatial effects at the receiver itself. This is the basis of ray tracing techniques'^*'^^ which
use site-specific measurements to predict path loss. Even current ray-tracing models focus
on single-channel measurements of power-delay profiles. This gives a macroscopic
description of the transmission amplitude versus delay but ignores local spatial
characteristics at the receiver.
Other models have also been proposed which describe the impact of the spatial
channel locally at the receiver but do not consider the surrounding environment. Salz and
Winters'"' examined the effects of a continuum of wavefronts with a nonzero DOA spread
on the crosscorrelations between array elements. As an example, the simple fading model
used in this research assumes the crosscorrelations are zero. Raleigh et al."" have also
112
proposed a time/space model to describe the effects of multipath on the spatial samples
from an array. There is plenty of room for additional local models which incorporate
space/time descriptions for an array-based receiver.
These models seem like a first tentative step towards a unified channel description.
Local receiver models need to be integrated into the larger-scale models of the
surrounding environment. This may dictate new, or at least more detailed, descriptions of
the processes which comprise the unified time/space channel.
APPENDIX A
THE OUTPUT S>fR PERFORMANCE SURFACE
This appendix gives some of the general aspects of the output SNR as a performance
surface. There are some important differences between the MSE and the SNR
performance surfaces. Some general qualities of the surface in weight-space are noted,
such as the contours and the gradient, along with the impact of misadjustment on the
multi-user processor.
The best way to get an intuitive feel for the SNR performance of an MMSE processor
is to examine the case of a single user in additive white Gaussian noise. The weight-space
diagram for a single-user case is shown in the figure below. The elliptical curves are the
customary constant MSE contours and the straight lines represent contours of constant
output SNR. Notice that the constant SNR contours are lines which pass through the
origin and only the slopes change with changing SNRo. As the weight vector moves away
from the MMSE weight vector, Wo, the output SNR decreases from the maximum output
SNROmax = NSNRi ( = 20 for this scenario). As the MSE increases the output SNR
decreases.
For a single user, two-weight array the relationship between SNRo and the weights is
easily shown through the relationship:
n»rn i w^Rw* w^MLM^w* v^Lv* _ _
SNRo+i= 3 ^ = — ^^ = -Y^r- (A-1)
(T,W w* a„w W* CT.V V*
113
114
Where L and M are the eigenvalue and eigenvector matrices of R respectively. The
Constant MSE.SNR Contours
0.5
-0.5
1
/ /sNRo=17.5
SNRo=14.o/
/^V
/ ^\>^0P^\\
SNRo=17^5,^-'-^^'^^ \
1^—^-3:^==^
SNRo=14.0
IZ^^^^^/^
SNRi=10
/
DOA=0 deg
-0.5
0.5
W(1)
Figure A. 1 Output contours for a two-element array with a single user. The straight lines
are constant contours of output SNR. The ellipses are constant MSE contours.
equation for the constant-value contour is v°Lv - a\{SNRo + l)v^v = .
For a single user in AWGN the eigenvalues are X,i = On, 'ki = P^ + d^. and the two
weights in the rotated coordinate system are related by the expression:
V, = ±v.
SNRo
SNRi-SNRo
(A.2)
which is a set of lines in two-dimensional, rotated weight-space. The SNR surface in
weight-space is concave. These relationships have been attributed to Winkler,'^* but the
original reference does not seem to be available. In the two-dimensional system shown in
lis
the figure above the major axis of the constant MSE ellipse is parallel to the noise-only
eigenvector and the ellipse minor axis is parallel to the signal and noise eigenvector. The
optimum MMSE weight vector Wo will be almost orthogonal to the noise-only
eigenvector.
The gradient of the output SNR is given by:
„ „.,_ 2SNRol , J SNRo.l
Vy,SNRo = „ „, u u' 1 w (A.3)
||wf L SNRi J
where u is the phase-shift vector defined in equation (1.4) and I is the identity matrix. It
might be argued that the norm of the gradient of SNRo is a measure of the local variations
of the SNRo function in weight-space. The Laplacian is the norm-squared of the gradient:
Vy,SNRof = ^^T^ [NSNRi - SNRo] (A.4)
||w|
Some things to note here are that the norm of the gradient is zero when the output SNR is
maximum ( = NSNRi) and that the gradient norm decreases as the weight vector norm
increases. It is possible to express output SNR in weight-space angles:
I T I
SNRo = SNRi i^]^-J^ = N ■ SNRi cos' a (A.5)
l|w||
where a is the angle between w and u in weight space. If equation (A.5) is substituted into
equation (A.4) the norm-squared of the gradient becomes:
W^SNRo
1 dSNRo^^
W da
(A.6)
This relationship is apparent in Figure (A. 1) above, variations in SNRo occur when the
weight vector is displaced in angle a from the phase-shift vector (which is collinear with
Wo in the figure).
116
For a two-user system with a weak user (1) and a strong user (2) the optimum weight
is almost collinear with the weaker user's phase-shift vector. Perturbations from the
steady-state weight vector, caused by the noisy gradient estimates of the LMS algorithm,
will have little effect on the weaker user since Oi (the angle between ui and Wo) is small.
Or, put another way, the difference term in the brackets of equation ( A.4) is small since
SNRo] = NSNRij, and the resulting variations are small. For a strong user, the difference in
the bracketed term may be large since SNR02 ~ SNRoi and the weight variations may cause
the large fluctuations in the strong user's output.
10
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BIOGRAPHICAL SKETCH
John Miller received his Bachelor of Science degree from the University of Rorida in
December, 1981. From February 1982 to April 1989 he worked at the Portable Products
Division of Motorola Inc. in Plantation, Florida. He returned to the University of Florida
in August 1989 and was awarded a Masters of Science degree in electrical engineering in
May 1991. Since this time he has been working towards a Ph.D.
129
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Scott L. Miller, Chairman
Associate Professor of Electrical
and Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosopl
Le6n W. Couch, II
Professor of Electrical and Computer
Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quaUty,
as a dissertation for the degree of Doctor of Philosophy.
John M. M. Anderson
Assistant Professor of Electrical
and Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fuUyjdg^uate, in scope and quality,
as a dissertation for the degree of Doctor of PhilosopfiyT
JoseyPnnc _
ProressQP^if Eleemcal and Computer
ingineenn|
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Ramesh Shrestha
Associate Professor of Civil
Engineering
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
August, 1996 r
/- Winfted M. PhilUps
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School
LD
1780
1996
SCIENCE
LIBRAJ«Y
UNIVERSITY OF FLORIDA
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